def _sqrt_symbolic_denest(a, b, r):
    """Given an expression, sqrt(a + b*sqrt(b)), return the denested
    expression or None.

    Algorithm:
    If r = ra + rb*sqrt(rr), try replacing sqrt(rr) in ``a`` with
    (y**2 - ra)/rb, and if the result is a quadratic, ca*y**2 + cb*y + cc, and
    (cb + b)**2 - 4*ca*cc is 0, then sqrt(a + b*sqrt(r)) can be rewritten as
    sqrt(ca*(sqrt(r) + (cb + b)/(2*ca))**2).

    Examples
    ========

    >>> from sympy.simplify.sqrtdenest import _sqrt_symbolic_denest, sqrtdenest
    >>> from sympy import sqrt, Symbol
    >>> from sympy.abc import x

    >>> a, b, r = 16 - 2*sqrt(29), 2, -10*sqrt(29) + 55
    >>> _sqrt_symbolic_denest(a, b, r)
    sqrt(-2*sqrt(29) + 11) + sqrt(5)

    If the expression is numeric, it will be simplified:

    >>> w = sqrt(sqrt(sqrt(3) + 1) + 1) + 1 + sqrt(2)
    >>> sqrtdenest(sqrt((w**2).expand()))
    1 + sqrt(2) + sqrt(1 + sqrt(1 + sqrt(3)))

    Otherwise, it will only be simplified if assumptions allow:

    >>> w = w.subs(sqrt(3), sqrt(x + 3))
    >>> sqrtdenest(sqrt((w**2).expand()))
    sqrt((sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2))**2)

    Notice that the argument of the sqrt is a square. If x is made positive
    then the sqrt of the square is resolved:

    >>> _.subs(x, Symbol('x', positive=True))
    sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2)
    """

    a, b, r = map(sympify, (a, b, r))
    rval = _sqrt_match(r)
    if not rval:
        return None
    ra, rb, rr = rval
    if rb:
        y = Dummy('y', positive=True)
        try:
            newa = Poly(a.subs(sqrt(rr), (y**2 - ra)/rb), y)
        except PolynomialError:
            return None
        if newa.degree() == 2:
            ca, cb, cc = newa.all_coeffs()
            cb += b
            if _mexpand(cb**2 - 4*ca*cc).equals(0):
                z = sqrt(ca*(sqrt(r) + cb/(2*ca))**2)
                if z.is_number:
                    z = _mexpand(Mul._from_args(z.as_content_primitive()))
                return z

def _sqrtdenest_rec(expr):
    """Helper that denests the square root of three or more surds.

    It returns the denested expression; if it cannot be denested it
    throws SqrtdenestStopIteration

    Algorithm: expr.base is in the extension Q_m = Q(sqrt(r_1),..,sqrt(r_k));
    split expr.base = a + b*sqrt(r_k), where `a` and `b` are on
    Q_(m-1) = Q(sqrt(r_1),..,sqrt(r_(k-1))); then a**2 - b**2*r_k is
    on Q_(m-1); denest sqrt(a**2 - b**2*r_k) and so on.
    See [1], section 6.

    Examples
    ========

    >>> from sympy import sqrt
    >>> from sympy.simplify.sqrtdenest import _sqrtdenest_rec
    >>> _sqrtdenest_rec(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
    -sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
    >>> w=-6*sqrt(55)-6*sqrt(35)-2*sqrt(22)-2*sqrt(14)+2*sqrt(77)+6*sqrt(10)+65
    >>> _sqrtdenest_rec(sqrt(w))
    -sqrt(11) - sqrt(7) + sqrt(2) + 3*sqrt(5)
    """
    from sympy.simplify.simplify import radsimp, split_surds, rad_rationalize
    if not expr.is_Pow:
        return sqrtdenest(expr)
    if expr.base < 0:
        return sqrt(-1)*_sqrtdenest_rec(sqrt(-expr.base))
    g, a, b = split_surds(expr.base)
    a = a*sqrt(g)
    if a < b:
        a, b = b, a
    c2 = _mexpand(a**2 - b**2)
    if len(c2.args) > 2:
        g, a1, b1 = split_surds(c2)
        a1 = a1*sqrt(g)
        if a1 < b1:
            a1, b1 = b1, a1
        c2_1 = _mexpand(a1**2 - b1**2)
        c_1 = _sqrtdenest_rec(sqrt(c2_1))
        d_1 = _sqrtdenest_rec(sqrt(a1 + c_1))
        num, den = rad_rationalize(b1, d_1)
        c = _mexpand(d_1/sqrt(2) + num/(den*sqrt(2)))
    else:
        c = _sqrtdenest1(sqrt(c2))

    if sqrt_depth(c) > 1:
        raise SqrtdenestStopIteration
    ac = a + c
    if len(ac.args) >= len(expr.args):
        if count_ops(ac) >= count_ops(expr.base):
            raise SqrtdenestStopIteration
    d = sqrtdenest(sqrt(ac))
    if sqrt_depth(d) > 1:
        raise SqrtdenestStopIteration
    num, den = rad_rationalize(b, d)
    r = d/sqrt(2) + num/(den*sqrt(2))
    r = radsimp(r)
    return _mexpand(r)

def _sqrtdenest1(expr, denester=True):
    """Return denested expr after denesting with simpler methods or, that
    failing, using the denester."""

    from sympy.simplify.simplify import radsimp

    if not is_sqrt(expr):
        return expr

    a = expr.base
    if a.is_Atom:
        return expr
    val = _sqrt_match(a)
    if not val:
        return expr

    a, b, r = val
    # try a quick numeric denesting
    d2 = _mexpand(a**2 - b**2*r)
    if d2.is_Rational:
        if d2.is_positive:
            z = _sqrt_numeric_denest(a, b, r, d2)
            if z is not None:
                return z
        else:
            # fourth root case
            # sqrtdenest(sqrt(3 + 2*sqrt(3))) =
            # sqrt(2)*3**(1/4)/2 + sqrt(2)*3**(3/4)/2
            dr2 = _mexpand(-d2*r)
            dr = sqrt(dr2)
            if dr.is_Rational:
                z = _sqrt_numeric_denest(_mexpand(b*r), a, r, dr2)
                if z is not None:
                    return z/root(r, 4)

    else:
        z = _sqrt_symbolic_denest(a, b, r)
        if z is not None:
            return z

    if not denester or not is_algebraic(expr):
        return expr

    res = sqrt_biquadratic_denest(expr, a, b, r, d2)
    if res:
        return res

    # now call to the denester
    av0 = [a, b, r, d2]
    z = _denester([radsimp(expr**2)], av0, 0, sqrt_depth(expr))[0]
    if av0[1] is None:
        return expr
    if z is not None:
        if sqrt_depth(z) == sqrt_depth(expr) and count_ops(z) > count_ops(expr):
            return expr
        return z
    return expr

def check_solutions(eq):
    """
    Determines whether solutions returned by diophantine() satisfy the original
    equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
    check_solutions_normal, check_solutions()
    """
    s = diophantine(eq)

    terms = factor_list(eq)[1]

    var = list(eq.free_symbols)
    var.sort(key=default_sort_key)

    okay = True

    while len(s) and okay:
        solution = s.pop()

        okay = False

        for term in terms:
            subeq = term[0]

            if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
                okay = True
                break

    return okay

def is_pell_transformation_ok(eq):
    """
    Test whether X*Y, X, or Y terms are present in the equation
    after transforming the equation using the transformation returned
    by transformation_to_pell(). If they are not present we are good.
    Moreover, coefficient of X**2 should be a divisor of coefficient of
    Y**2 and the constant term.
    """
    A, B = transformation_to_DN(eq)
    u = (A*Matrix([X, Y]) + B)[0]
    v = (A*Matrix([X, Y]) + B)[1]
    simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())

    coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])

    for term in [X*Y, X, Y]:
        if term in coeff.keys():
            return False

    for term in [X**2, Y**2, Integer(1)]:
        if term not in coeff.keys():
            coeff[term] = Integer(0)

    if coeff[X**2] != 0:
        return isinstance(S(coeff[Y**2])/coeff[X**2], Integer) and isinstance(S(coeff[Integer(1)])/coeff[X**2], Integer)

    return True

def _minpoly_cos(ex, x):
    """
    Returns the minimal polynomial of ``cos(ex)``
    see http://mathworld.wolfram.com/TrigonometryAngles.html
    """
    from sympy import sqrt
    c, a = ex.args[0].as_coeff_Mul()
    if a is pi:
        if c.is_rational:
            if c.p == 1:
                if c.q == 7:
                    return 8*x**3 - 4*x**2 - 4*x + 1
                if c.q == 9:
                    return 8*x**3 - 6*x + 1
            elif c.p == 2:
                q = sympify(c.q)
                if q.is_prime:
                    s = _minpoly_sin(ex, x)
                    return _mexpand(s.subs({x:sqrt((1 - x)/2)}))

            # for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
            n = int(c.q)
            a = dup_chebyshevt(n, ZZ)
            a = [x**(n - i)*a[i] for i in range(n + 1)]
            r = Add(*a) - (-1)**c.p
            _, factors = factor_list(r)
            res = _choose_factor(factors, x, ex)
            return res

    raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)

def _sqrtdenest0(expr):
    """Returns expr after denesting its arguments."""

    if is_sqrt(expr):
        n, d = expr.as_numer_denom()
        if d is S.One:  # n is a square root
            if n.base.is_Add:
                args = sorted(n.base.args, key=default_sort_key)
                if len(args) > 2 and all((x**2).is_Integer for x in args):
                    try:
                        return _sqrtdenest_rec(n)
                    except SqrtdenestStopIteration:
                        pass
                expr = sqrt(_mexpand(Add(*[_sqrtdenest0(x) for x in args])))
            return _sqrtdenest1(expr)
        else:
            n, d = [_sqrtdenest0(i) for i in (n, d)]
            return n/d

    if isinstance(expr, Add):
        cs = []
        args = []
        for arg in expr.args:
            c, a = arg.as_coeff_Mul()
            cs.append(c)
            args.append(a)

        if all(c.is_Rational for c in cs) and all(is_sqrt(arg) for arg in args):
            return _sqrt_ratcomb(cs, args)

    if isinstance(expr, Expr):
        args = expr.args
        if args:
            return expr.func(*[_sqrtdenest0(a) for a in args])
    return expr

 def find(a):
     n = len(a)
     for i in range(n - 1):
         for j in range(i + 1, n):
             s1 = a[i].base
             s2 = a[j].base
             p = _mexpand(s1 * s2)
             s = sqrtdenest(sqrt(p))
             if s != sqrt(p):
                 return s, i, j

def nthroot(expr, n, max_len=4, prec=15):
    """
    compute a real nth-root of a sum of surds

    Parameters
    ==========

    expr : sum of surds
    n : integer
    max_len : maximum number of surds passed as constants to ``nsimplify``

    Algorithm
    =========

    First ``nsimplify`` is used to get a candidate root; if it is not a
    root the minimal polynomial is computed; the answer is one of its
    roots.

    Examples
    ========

    >>> from sympy.simplify.simplify import nthroot
    >>> from sympy import Rational, sqrt
    >>> nthroot(90 + 34*sqrt(7), 3)
    sqrt(7) + 3

    """
    expr = sympify(expr)
    n = sympify(n)
    p = expr**Rational(1, n)
    if not n.is_integer:
        return p
    if not _is_sum_surds(expr):
        return p
    surds = []
    coeff_muls = [x.as_coeff_Mul() for x in expr.args]
    for x, y in coeff_muls:
        if not x.is_rational:
            return p
        if y is S.One:
            continue
        if not (y.is_Pow and y.exp == S.Half and y.base.is_integer):
            return p
        surds.append(y)
    surds.sort()
    surds = surds[:max_len]
    if expr < 0 and n % 2 == 1:
        p = (-expr)**Rational(1, n)
        a = nsimplify(p, constants=surds)
        res = a if _mexpand(a**n) == _mexpand(-expr) else p
        return -res
    a = nsimplify(p, constants=surds)
    if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr):
        return _mexpand(a)
    expr = _nthroot_solve(expr, n, prec)
    if expr is None:
        return p
    return expr

def rad_rationalize(num, den):
    """
    Rationalize num/den by removing square roots in the denominator;
    num and den are sum of terms whose squares are rationals

    Examples
    ========

    >>> from sympy import sqrt
    >>> from sympy.simplify.radsimp import rad_rationalize
    >>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3)
    (-sqrt(3) + sqrt(6)/3, -7/9)
    """
    if not den.is_Add:
        return num, den
    g, a, b = split_surds(den)
    a = a*sqrt(g)
    num = _mexpand((a - b)*num)
    den = _mexpand(a**2 - b**2)
    return rad_rationalize(num, den)

def is_normal_transformation_ok(eq):
    A = transformation_to_normal(eq)
    X, Y, Z = A*Matrix([x, y, z])
    simplified = _mexpand(Subs(eq, (x, y, z), (X, Y, Z)).doit())

    coeff = dict([reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args])
    for term in [X*Y, Y*Z, X*Z]:
        if term in coeff.keys():
            return False

    return True

def _lambert(eq, x):
    """
    Given an expression assumed to be in the form
        ``F(X, a..f) = a*log(b*X + c) + d*X + f = 0``
    where X = g(x) and x = g^-1(X), return the Lambert solution if possible:
        ``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``.
    """
    eq = _mexpand(expand_log(eq))
    mainlog = _mostfunc(eq, log, x)
    if not mainlog:
        return []  # violated assumptions
    other = eq.subs(mainlog, 0)
    if (-other).func is log:
        eq = (eq - other).subs(mainlog, mainlog.args[0])
        mainlog = mainlog.args[0]
        if mainlog.func is not log:
            return []  # violated assumptions
        other = -(-other).args[0]
        eq += other
    if not x in other.free_symbols:
        return [] # violated assumptions
    d, f, X2 = _linab(other, x)
    logterm = collect(eq - other, mainlog)
    a = logterm.as_coefficient(mainlog)
    if a is None or x in a.free_symbols:
        return []  # violated assumptions
    logarg = mainlog.args[0]
    b, c, X1 = _linab(logarg, x)
    if X1 != X2:
        return []  # violated assumptions

    u = Dummy('rhs')
    sol = []
    # check only real solutions:
    for k in [-1, 0]:
        l = LambertW(d/(a*b)*exp(c*d/a/b)*exp(-f/a), k)
        # if W's arg is between -1/e and 0 there is
        # a -1 branch real solution, too.
        if k and not l.is_real:
            continue
        rhs = -c/b + (a/d)*l

        solns = solve(X1 - u, x)
        for i, tmp in enumerate(solns):
            solns[i] = tmp.subs(u, rhs)
            sol.append(solns[i])
    return sol

def _sqrtdenest0(expr):
    """Returns expr after denesting its arguments."""

    if is_sqrt(expr):
        n, d = expr.as_numer_denom()
        if d is S.One:  # n is a square root
            if n.base.is_Add:
                args = sorted(n.base.args, key=default_sort_key)
                if len(args) > 2 and all((x**2).is_Integer for x in args):
                    try:
                        return _sqrtdenest_rec(n)
                    except SqrtdenestStopIteration:
                        pass
                expr = sqrt(_mexpand(Add(*[_sqrtdenest0(x) for x in args])))
            return _sqrtdenest1(expr)
        else:
            n, d = [_sqrtdenest0(i) for i in (n, d)]
            return n/d
    if isinstance(expr, Expr):
        args = expr.args
        if args:
            return expr.func(*[_sqrtdenest0(a) for a in args])
    return expr

def _nthroot_solve(p, n, prec):
    """
     helper function for ``nthroot``
     It denests ``p**Rational(1, n)`` using its minimal polynomial
    """
    from sympy.polys.numberfields import _minimal_polynomial_sq
    from sympy.solvers import solve
    while n % 2 == 0:
        p = sqrtdenest(sqrt(p))
        n = n // 2
    if n == 1:
        return p
    pn = p**Rational(1, n)
    x = Symbol('x')
    f = _minimal_polynomial_sq(p, n, x)
    if f is None:
        return None
    sols = solve(f, x)
    for sol in sols:
        if abs(sol - pn).n() < 1./10**prec:
            sol = sqrtdenest(sol)
            if _mexpand(sol**n) == p:
                return sol

def test_mexpand():
    from sympy.abc import x
    assert _mexpand(None) is None
    assert _mexpand(1) is S.One
    assert _mexpand(x*(x + 1)**2) == (x*(x + 1)**2).expand()

def _separate_sq(p):
    """
    helper function for ``_minimal_polynomial_sq``

    It selects a rational ``g`` such that the polynomial ``p``
    consists of a sum of terms whose surds squared have gcd equal to ``g``
    and a sum of terms with surds squared prime with ``g``;
    then it takes the field norm to eliminate ``sqrt(g)``

    See simplify.simplify.split_surds and polytools.sqf_norm.

    Examples
    ========

    >>> from sympy import sqrt
    >>> from sympy.abc import x
    >>> from sympy.polys.numberfields import _separate_sq
    >>> p= -x + sqrt(2) + sqrt(3) + sqrt(7)
    >>> p = _separate_sq(p); p
    -x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8
    >>> p = _separate_sq(p); p
    -x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20
    >>> p = _separate_sq(p); p
    -x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400

    """
    from sympy.utilities.iterables import sift
    def is_sqrt(expr):
        return expr.is_Pow and expr.exp is S.Half
    # p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)]
    a = []
    for y in p.args:
        if not y.is_Mul:
            if is_sqrt(y):
                a.append((S.One, y**2))
            elif y.is_Atom:
                a.append((y, S.One))
            elif y.is_Pow and y.exp.is_integer:
                a.append((y, S.One))
            else:
                raise NotImplementedError
            continue
        T, F = sift(y.args, is_sqrt, binary=True)
        a.append((Mul(*F), Mul(*T)**2))
    a.sort(key=lambda z: z[1])
    if a[-1][1] is S.One:
        # there are no surds
        return p
    surds = [z for y, z in a]
    for i in range(len(surds)):
        if surds[i] != 1:
            break
    g, b1, b2 = _split_gcd(*surds[i:])
    a1 = []
    a2 = []
    for y, z in a:
        if z in b1:
            a1.append(y*z**S.Half)
        else:
            a2.append(y*z**S.Half)
    p1 = Add(*a1)
    p2 = Add(*a2)
    p = _mexpand(p1**2) - _mexpand(p2**2)
    return p

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic element.

    Parameters
    ==========

    ex : algebraic element expression
    x : independent variable of the minimal polynomial

    Options
    =======

    compose : if ``True`` ``_minpoly_compose`` is used, if ``False`` the ``groebner`` algorithm
    polys : if ``True`` returns a ``Poly`` object
    domain : ground domain

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    If no ground domain is given, it will be generated automatically from the expression.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve, QQ
    >>> from sympy.abc import x, y

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
    x - sqrt(2)
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3
    >>> minimal_polynomial(sqrt(y), x)
    x**2 - y

    """
    from sympy.polys.polytools import degree
    from sympy.polys.domains import FractionField
    from sympy.core.basic import preorder_traversal

    compose = args.get('compose', True)
    polys = args.get('polys', False)
    dom = args.get('domain', None)

    ex = sympify(ex)
    if ex.is_number:
        # not sure if it's always needed but try it for numbers (issue 8354)
        ex = _mexpand(ex, recursive=True)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if not dom:
        dom = FractionField(QQ, list(ex.free_symbols)) if ex.free_symbols else QQ
    if hasattr(dom, 'symbols') and x in dom.symbols:
        raise GeneratorsError("the variable %s is an element of the ground domain %s" % (x, dom))

    if compose:
        result = _minpoly_compose(ex, x, dom)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c.is_negative:
            result = expand_mul(-result)
        return cls(result, x, field=True) if polys else result.collect(x)

    if not dom.is_QQ:
        raise NotImplementedError("groebner method only works for QQ")

    result = _minpoly_groebner(ex, x, cls)
    return cls(result, x, field=True) if polys else result.collect(x)

 def ok(f, v, c):
     new = _mexpand(f.subs(v, c))
     free = new.free_symbols
     return None if (x in free or y in free) else new

def test_factor_nc():
    x, y = symbols('x,y')
    k = symbols('k', integer=True)
    n, m, o = symbols('n,m,o', commutative=False)

    # mul and multinomial expansion is needed
    from sympy.core.function import _mexpand
    e = x*(1 + y)**2
    assert _mexpand(e) == x + x*2*y + x*y**2

    def factor_nc_test(e):
        ex = _mexpand(e)
        assert ex.is_Add
        f = factor_nc(ex)
        assert not f.is_Add and _mexpand(f) == ex

    factor_nc_test(x*(1 + y))
    factor_nc_test(n*(x + 1))
    factor_nc_test(n*(x + m))
    factor_nc_test((x + m)*n)
    factor_nc_test(n*m*(x*o + n*o*m)*n)
    s = Sum(x, (x, 1, 2))
    factor_nc_test(x*(1 + s))
    factor_nc_test(x*(1 + s)*s)
    factor_nc_test(x*(1 + sin(s)))
    factor_nc_test((1 + n)**2)

    factor_nc_test((x + n)*(x + m)*(x + y))
    factor_nc_test(x*(n*m + 1))
    factor_nc_test(x*(n*m + x))
    factor_nc_test(x*(x*n*m + 1))
    factor_nc_test(x*n*(x*m + 1))
    factor_nc_test(x*(m*n + x*n*m))
    factor_nc_test(n*(1 - m)*n**2)

    factor_nc_test((n + m)**2)
    factor_nc_test((n - m)*(n + m)**2)
    factor_nc_test((n + m)**2*(n - m))
    factor_nc_test((m - n)*(n + m)**2*(n - m))

    assert factor_nc(n*(n + n*m)) == n**2*(1 + m)
    assert factor_nc(m*(m*n + n*m*n**2)) == m*(m + n*m*n)*n
    eq = m*sin(n) - sin(n)*m
    assert factor_nc(eq) == eq

    # for coverage:
    from sympy.physics.secondquant import Commutator
    from sympy import factor
    eq = 1 + x*Commutator(m, n)
    assert factor_nc(eq) == eq
    eq = x*Commutator(m, n) + x*Commutator(m, o)*Commutator(m, n)
    assert factor(eq) == x*(1 + Commutator(m, o))*Commutator(m, n)

    # issue 6534
    assert (2*n + 2*m).factor() == 2*(n + m)

    # issue 6701
    assert factor_nc(n**k + n**(k + 1)) == n**k*(1 + n)
    assert factor_nc((m*n)**k + (m*n)**(k + 1)) == (1 + m*n)*(m*n)**k

    # issue 6918
    assert factor_nc(-n*(2*x**2 + 2*x)) == -2*n*x*(x + 1)

def bivariate_type(f, x, y, **kwargs):
    """Given an expression, f, 3 tests will be done to see what type
    of composite bivariate it might be, options for u(x, y) are::

        x*y
        x+y
        x*y+x
        x*y+y

    If it matches one of these types, ``u(x, y)``, ``P(u)`` and dummy
    variable ``u`` will be returned. Solving ``P(u)`` for ``u`` and
    equating the solutions to ``u(x, y)`` and then solving for ``x`` or
    ``y`` is equivalent to solving the original expression for ``x`` or
    ``y``. If ``x`` and ``y`` represent two functions in the same
    variable, e.g. ``x = g(t)`` and ``y = h(t)``, then if ``u(x, y) - p``
    can be solved for ``t`` then these represent the solutions to
    ``P(u) = 0`` when ``p`` are the solutions of ``P(u) = 0``.

    Only positive values of ``u`` are considered.

    Examples
    ========

    >>> from sympy.solvers.solvers import solve
    >>> from sympy.solvers.bivariate import bivariate_type
    >>> from sympy.abc import x, y
    >>> eq = (x**2 - 3).subs(x, x + y)
    >>> bivariate_type(eq, x, y)
    (x + y, _u**2 - 3, _u)
    >>> uxy, pu, u = _
    >>> usol = solve(pu, u); usol
    [sqrt(3)]
    >>> [solve(uxy - s) for s in solve(pu, u)]
    [[{x: -y + sqrt(3)}]]
    >>> all(eq.subs(s).equals(0) for sol in _ for s in sol)
    True

    """

    u = Dummy('u', positive=True)

    if kwargs.pop('first', True):
        p = Poly(f, x, y)
        f = p.as_expr()
        _x = Dummy()
        _y = Dummy()
        rv = bivariate_type(Poly(f.subs({x: _x, y: _y}), _x, _y), _x, _y, first=False)
        if rv:
            reps = {_x: x, _y: y}
            return rv[0].xreplace(reps), rv[1].xreplace(reps), rv[2]
        return

    p = f
    f = p.as_expr()

    # f(x*y)
    args = Add.make_args(p.as_expr())
    new = []
    for a in args:
        a = _mexpand(a.subs(x, u/y))
        free = a.free_symbols
        if x in free or y in free:
            break
        new.append(a)
    else:
        return x*y, Add(*new), u

    def ok(f, v, c):
        new = _mexpand(f.subs(v, c))
        free = new.free_symbols
        return None if (x in free or y in free) else new

    # f(a*x + b*y)
    new = []
    d = p.degree(x)
    if p.degree(y) == d:
        a = root(p.coeff_monomial(x**d), d)
        b = root(p.coeff_monomial(y**d), d)
        new = ok(f, x, (u - b*y)/a)
        if new is not None:
            return a*x + b*y, new, u

    # f(a*x*y + b*y)
    new = []
    d = p.degree(x)
    if p.degree(y) == d:
        for itry in range(2):
            a = root(p.coeff_monomial(x**d*y**d), d)
            b = root(p.coeff_monomial(y**d), d)
            new = ok(f, x, (u - b*y)/a/y)
            if new is not None:
                return a*x*y + b*y, new, u
            x, y = y, x