def test_factor_terms():
    A = Symbol('A', commutative=False)
    assert factor_terms(9*(x + x*y + 1) + (3*x + 3)**(2 + 2*x)) == \
        9*x*y + 9*x + _keep_coeff(S(3), x + 1)**_keep_coeff(S(2), x + 1) + 9
    assert factor_terms(9*(x + x*y + 1) + (3)**(2 + 2*x)) == \
        _keep_coeff(S(9), 3**(2*x) + x*y + x + 1)
    assert factor_terms(3**(2 + 2*x) + a*3**(2 + 2*x)) == \
        9*3**(2*x)*(a + 1)
    assert factor_terms(x + x*A) == \
        x*(1 + A)
    assert factor_terms(sin(x + x*A)) == \
        sin(x*(1 + A))
    assert factor_terms((3*x + 3)**((2 + 2*x)/3)) == \
        _keep_coeff(S(3), x + 1)**_keep_coeff(S(2)/3, x + 1)
    assert factor_terms(x + (x*y + x)**(3*x + 3)) == \
        x + (x*(y + 1))**_keep_coeff(S(3), x + 1)
    assert factor_terms(a*(x + x*y) + b*(x*2 + y*x*2)) == \
        x*(a + 2*b)*(y + 1)
    i = Integral(x, (x, 0, oo))
    assert factor_terms(i) == i

    # check radical extraction
    eq = sqrt(2) + sqrt(10)
    assert factor_terms(eq) == eq
    assert factor_terms(eq, radical=True) == sqrt(2)*(1 + sqrt(5))
    eq = root(-6, 3) + root(6, 3)
    assert factor_terms(eq, radical=True) == 6**(S(1)/3)*(1 + (-1)**(S(1)/3))

    eq = [x + x*y]
    ans = [x*(y + 1)]
    for c in [list, tuple, set]:
        assert factor_terms(c(eq)) == c(ans)
    assert factor_terms(Tuple(x + x*y)) == Tuple(x*(y + 1))
    assert factor_terms(Interval(0, 1)) == Interval(0, 1)
    e = 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=False) == 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=True) == sqrt(2)/sqrt(a + 2)

    eq = x/(x + 1/x) + 1/(x**2 + 1)
    assert factor_terms(eq, fraction=False) == eq
    assert factor_terms(eq, fraction=True) == 1

    assert factor_terms((1/(x**3 + x**2) + 2/x**2)*y) == \
        y*(2 + 1/(x + 1))/x**2

    # if not True, then processesing for this in factor_terms is not necessary
    assert gcd_terms(-x - y) == -x - y
    assert factor_terms(-x - y) == Mul(-1, x + y, evaluate=False)

    # if not True, then "special" processesing in factor_terms is not necessary
    assert gcd_terms(exp(Mul(-1, x + 1))) == exp(-x - 1)
    e = exp(-x - 2) + x
    assert factor_terms(e) == exp(Mul(-1, x + 2, evaluate=False)) + x
    assert factor_terms(e, sign=False) == e
    assert factor_terms(exp(-4*x - 2) - x) == -x + exp(Mul(-2, 2*x + 1, evaluate=False))

def test_gcd_terms():
    f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + (2*x + 2)*(x + 6)/(5*x**2 + 5)

    assert _gcd_terms(f) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)
    assert _gcd_terms(Add.make_args(f)) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)

    assert gcd_terms(f) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))
    assert gcd_terms(Add.make_args(f)) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))

    assert gcd_terms((2*x + 2)**3 + (2*x + 2)**2) == 4*(x + 1)**2*(2*x + 3)

    assert gcd_terms(0) == 0
    assert gcd_terms(1) == 1
    assert gcd_terms(x) == x
    assert gcd_terms(2 + 2*x) == Mul(2, 1 + x, evaluate=False)
    arg = x*(2*x + 4*y)
    garg = 2*x*(x + 2*y)
    assert gcd_terms(arg) == garg
    assert gcd_terms(sin(arg)) == sin(garg)

    # issue 3040-like
    alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
    a = alpha**2 - alpha*x**2 + alpha + x**3 - x*(alpha + 1)
    rep = (alpha, (1 + sqrt(5))/2 + alpha1*x + alpha2*x**2 + alpha3*x**3)
    s = (a/(x - alpha)).subs(*rep).series(x, 0, 1)
    assert simplify(collect(s, x)) == -sqrt(5)/2 - S(3)/2 + O(x)

    # issue 2818
    assert _gcd_terms([S.Zero, S.Zero]) == (0, 0, 1)
    assert _gcd_terms([2*x + 4]) == (2, x + 2, 1)

def test_gcd_terms():
    f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + (2*x + 2)*(x + 6)/(5*x**2 + 5)

    assert _gcd_terms(f) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)
    assert _gcd_terms(Add.make_args(f)) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)

    assert gcd_terms(f) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))
    assert gcd_terms(Add.make_args(f)) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))

    assert gcd_terms(0) == 0
    assert gcd_terms(1) == 1
    assert gcd_terms(x) == x

def test_gcd_terms():
    f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + (2*x + 2)*(x + 6)/(5*x**2 + 5)

    assert _gcd_terms(f) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)
    assert _gcd_terms(Add.make_args(f)) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)

    assert gcd_terms(f) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))
    assert gcd_terms(Add.make_args(f)) == (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))

    assert gcd_terms((2*x + 2)**3 + (2*x + 2)**2) == 4*(x + 1)**2*(2*x + 3)

    assert gcd_terms(0) == 0
    assert gcd_terms(1) == 1
    assert gcd_terms(x) == x
    assert gcd_terms(2 + 2*x) == Mul(2, 1 + x, evaluate=False)
    arg = x*(2*x + 4*y)
    garg = 2*x*(x + 2*y)
    assert gcd_terms(arg) == garg
    assert gcd_terms(sin(arg)) == sin(garg)

    def _together(expr):
        if isinstance(expr, Basic):
            if expr.is_Atom or (expr.is_Function and not deep):
                return expr
            elif expr.is_Add:
                return gcd_terms(map(_together, Add.make_args(expr)))
            elif expr.is_Pow:
                base = _together(expr.base)

                if deep:
                    exp = _together(expr.exp)
                else:
                    exp = expr.exp

                return expr.__class__(base, exp)
            else:
                return expr.__class__(*[ _together(arg) for arg in expr.args ])
        elif hasattr(expr, '__iter__'):
            return expr.__class__([ _together(ex) for ex in expr ])

        return expr

    def eval(cls, p, q):
        from sympy.core.add import Add
        from sympy.core.mul import Mul
        from sympy.core.singleton import S
        from sympy.core.exprtools import gcd_terms
        from sympy.polys.polytools import gcd

        def doit(p, q):
            """Try to return p % q if both are numbers or +/-p is known
            to be less than q.
            """

            if p == q or p == -q or p.is_Pow and p.exp.is_Integer and p.base == q:
                return S.Zero

            if p.is_Number and q.is_Number:
                return (p % q)

            # by ratio
            r = p/q
            try:
                d = int(r)
            except TypeError:
                pass
            else:
                if type(d) is int:
                    rv = p - d*q
                    if (rv*q < 0) is True:
                        rv += q
                    return rv

            # by differencec
            d = p - q
            if d.is_negative:
                if q.is_negative:
                    return d
                elif q.is_positive:
                    return p

        rv = doit(p, q)
        if rv is not None:
            return rv

        # denest
        if p.func is cls:
            # easy
            qinner = p.args[1]
            if qinner == q:
                return p
            # XXX other possibilities?

        # extract gcd; any further simplification should be done by the user
        G = gcd(p, q)
        if G is not S.One:
            p, q = [
                gcd_terms(i/G, clear=False, fraction=False) for i in (p, q)]
        pwas, qwas = p, q

        # simplify terms
        # (x + y + 2) % x -> Mod(y + 2, x)
        if p.is_Add:
            args = []
            for i in p.args:
                a = cls(i, q)
                if a.count(cls) > i.count(cls):
                    args.append(i)
                else:
                    args.append(a)
            if args != list(p.args):
                p = Add(*args)

        else:
            # handle coefficients if they are not Rational
            # since those are not handled by factor_terms
            # e.g. Mod(.6*x, .3*y) -> 0.3*Mod(2*x, y)
            cp, p = p.as_coeff_Mul()
            cq, q = q.as_coeff_Mul()
            ok = False
            if not cp.is_Rational or not cq.is_Rational:
                r = cp % cq
                if r == 0:
                    G *= cq
                    p *= int(cp/cq)
                    ok = True
            if not ok:
                p = cp*p
                q = cq*q

        # simple -1 extraction
        if p.could_extract_minus_sign() and q.could_extract_minus_sign():
            G, p, q = [-i for i in (G, p, q)]

        # check again to see if p and q can now be handled as numbers
        rv = doit(p, q)
        if rv is not None:
            return rv*G

        # put 1.0 from G on inside
        if G.is_Float and G == 1:
            p *= G
#.........这里部分代码省略.........

def test_gcd_terms():
    f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + \
        (2*x + 2)*(x + 6)/(5*x**2 + 5)

    assert _gcd_terms(f) == ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)
    assert _gcd_terms(Add.make_args(f)) == \
        ((S(6)/5)*((1 + x)/(1 + x**2)), 5 + x, 1)

    newf = (S(6)/5)*((1 + x)*(5 + x)/(1 + x**2))
    assert gcd_terms(f) == newf
    args = Add.make_args(f)
    # non-Basic sequences of terms treated as terms of Add
    assert gcd_terms(list(args)) == newf
    assert gcd_terms(tuple(args)) == newf
    assert gcd_terms(set(args)) == newf
    # but a Basic sequence is treated as a container
    assert gcd_terms(Tuple(*args)) != newf
    assert gcd_terms(Basic(Tuple(1, 3*y + 3*x*y), Tuple(1, 3))) == \
        Basic((1, 3*y*(x + 1)), (1, 3))
    # but we shouldn't change keys of a dictionary or some may be lost
    assert gcd_terms(Dict((x*(1 + y), 2), (x + x*y, y + x*y))) == \
        Dict({x*(y + 1): 2, x + x*y: y*(1 + x)})

    assert gcd_terms((2*x + 2)**3 + (2*x + 2)**2) == 4*(x + 1)**2*(2*x + 3)

    assert gcd_terms(0) == 0
    assert gcd_terms(1) == 1
    assert gcd_terms(x) == x
    assert gcd_terms(2 + 2*x) == Mul(2, 1 + x, evaluate=False)
    arg = x*(2*x + 4*y)
    garg = 2*x*(x + 2*y)
    assert gcd_terms(arg) == garg
    assert gcd_terms(sin(arg)) == sin(garg)

    # issue 3040-like
    alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
    a = alpha**2 - alpha*x**2 + alpha + x**3 - x*(alpha + 1)
    rep = (alpha, (1 + sqrt(5))/2 + alpha1*x + alpha2*x**2 + alpha3*x**3)
    s = (a/(x - alpha)).subs(*rep).series(x, 0, 1)
    assert simplify(collect(s, x)) == -sqrt(5)/2 - S(3)/2 + O(x)

    # issue 2818
    assert _gcd_terms([S.Zero, S.Zero]) == (0, 0, 1)
    assert _gcd_terms([2*x + 4]) == (2, x + 2, 1)

    eq = x/(x + 1/x)
    assert gcd_terms(eq, fraction=False) == eq

    def eval(cls, p, q):
        from sympy.core.add import Add
        from sympy.core.mul import Mul
        from sympy.core.singleton import S
        from sympy.core.exprtools import gcd_terms
        from sympy.polys.polytools import gcd

        def doit(p, q):
            """Try to return p % q if both are numbers or +/-p is known
            to be less than or equal q.
            """

            if q == S.Zero:
                raise ZeroDivisionError("Modulo by zero")
            if p.is_infinite or q.is_infinite or p is nan or q is nan:
                return nan
            if p == S.Zero or p == q or p == -q or (p.is_integer and q == 1):
                return S.Zero

            if q.is_Number:
                if p.is_Number:
                    return p%q
                if q == 2:
                    if p.is_even:
                        return S.Zero
                    elif p.is_odd:
                        return S.One

            if hasattr(p, '_eval_Mod'):
                rv = getattr(p, '_eval_Mod')(q)
                if rv is not None:
                    return rv

            # by ratio
            r = p/q
            try:
                d = int(r)
            except TypeError:
                pass
            else:
                if isinstance(d, integer_types):
                    rv = p - d*q
                    if (rv*q < 0) == True:
                        rv += q
                    return rv

            # by difference
            # -2|q| < p < 2|q|
            d = abs(p)
            for _ in range(2):
                d -= abs(q)
                if d.is_negative:
                    if q.is_positive:
                        if p.is_positive:
                            return d + q
                        elif p.is_negative:
                            return -d
                    elif q.is_negative:
                        if p.is_positive:
                            return d
                        elif p.is_negative:
                            return -d + q
                    break

        rv = doit(p, q)
        if rv is not None:
            return rv

        # denest
        if isinstance(p, cls):
            qinner = p.args[1]
            if qinner % q == 0:
                return cls(p.args[0], q)
            elif (qinner*(q - qinner)).is_nonnegative:
                # |qinner| < |q| and have same sign
                return p
        elif isinstance(-p, cls):
            qinner = (-p).args[1]
            if qinner % q == 0:
                return cls(-(-p).args[0], q)
            elif (qinner*(q + qinner)).is_nonpositive:
                # |qinner| < |q| and have different sign
                return p
        elif isinstance(p, Add):
            # separating into modulus and non modulus
            both_l = non_mod_l, mod_l = [], []
            for arg in p.args:
                both_l[isinstance(arg, cls)].append(arg)
            # if q same for all
            if mod_l and all(inner.args[1] == q for inner in mod_l):
                net = Add(*non_mod_l) + Add(*[i.args[0] for i in mod_l])
                return cls(net, q)

        elif isinstance(p, Mul):
            # separating into modulus and non modulus
            both_l = non_mod_l, mod_l = [], []
            for arg in p.args:
                both_l[isinstance(arg, cls)].append(arg)

            if mod_l and all(inner.args[1] == q for inner in mod_l):
#.........这里部分代码省略.........

#.........这里部分代码省略.........
        elif d.is_Pow and (d.exp.is_integer or d.base.is_positive):
            # (1/d**i) = (1/d)**i
            return handle(1/d.base)**d.exp

        if not (d.is_Add or ispow2(d)):
            return 1/d.func(*[handle(a) for a in d.args])

        # handle 1/d treating d as an Add (though it may not be)

        keep = True  # keep changes that are made

        # flatten it and collect radicals after checking for special
        # conditions
        d = _mexpand(d)

        # did it change?
        if d.is_Atom:
            return 1/d

        # is it a number that might be handled easily?
        if d.is_number:
            _d = nsimplify(d)
            if _d.is_Number and _d.equals(d):
                return 1/_d

        while True:
            # collect similar terms
            collected = defaultdict(list)
            for m in Add.make_args(d):  # d might have become non-Add
                p2 = []
                other = []
                for i in Mul.make_args(m):
                    if ispow2(i, log2=True):
                        p2.append(i.base if i.exp is S.Half else i.base**(2*i.exp))
                    elif i is S.ImaginaryUnit:
                        p2.append(S.NegativeOne)
                    else:
                        other.append(i)
                collected[tuple(ordered(p2))].append(Mul(*other))
            rterms = list(ordered(list(collected.items())))
            rterms = [(Mul(*i), Add(*j)) for i, j in rterms]
            nrad = len(rterms) - (1 if rterms[0][0] is S.One else 0)
            if nrad < 1:
                break
            elif nrad > max_terms:
                # there may have been invalid operations leading to this point
                # so don't keep changes, e.g. this expression is troublesome
                # in collecting terms so as not to raise the issue of 2834:
                # r = sqrt(sqrt(5) + 5)
                # eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r)
                keep = False
                break
            if len(rterms) > 4:
                # in general, only 4 terms can be removed with repeated squaring
                # but other considerations can guide selection of radical terms
                # so that radicals are removed
                if all([x.is_Integer and (y**2).is_Rational for x, y in rterms]):
                    nd, d = rad_rationalize(S.One, Add._from_args(
                        [sqrt(x)*y for x, y in rterms]))
                    n *= nd
                else:
                    # is there anything else that might be attempted?
                    keep = False
                break
            from sympy.simplify.powsimp import powsimp, powdenest

            num = powsimp(_num(rterms))
            n *= num
            d *= num
            d = powdenest(_mexpand(d), force=symbolic)
            if d.is_Atom:
                break

        if not keep:
            return expr
        return _unevaluated_Mul(n, 1/d)

    coeff, expr = expr.as_coeff_Add()
    expr = expr.normal()
    old = fraction(expr)
    n, d = fraction(handle(expr))
    if old != (n, d):
        if not d.is_Atom:
            was = (n, d)
            n = signsimp(n, evaluate=False)
            d = signsimp(d, evaluate=False)
            u = Factors(_unevaluated_Mul(n, 1/d))
            u = _unevaluated_Mul(*[k**v for k, v in u.factors.items()])
            n, d = fraction(u)
            if old == (n, d):
                n, d = was
        n = expand_mul(n)
        if d.is_Number or d.is_Add:
            n2, d2 = fraction(gcd_terms(_unevaluated_Mul(n, 1/d)))
            if d2.is_Number or (d2.count_ops() <= d.count_ops()):
                n, d = [signsimp(i) for i in (n2, d2)]
                if n.is_Mul and n.args[0].is_Number:
                    n = n.func(*n.args)

    return coeff + _unevaluated_Mul(n, 1/d)