3.1.25 \(\int \frac {1}{(a+b x^3)^2 (c+d x^3)} \, dx\) [25]

Optimal. Leaf size=346 \[ \frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}-\frac {b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2} \]

[Out]

1/3*b*x/a/(-a*d+b*c)/(b*x^3+a)+1/9*b^(2/3)*(-5*a*d+2*b*c)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/(-a*d+b*c)^2+1/3*d^(5/
3)*ln(c^(1/3)+d^(1/3)*x)/c^(2/3)/(-a*d+b*c)^2-1/18*b^(2/3)*(-5*a*d+2*b*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)
*x^2)/a^(5/3)/(-a*d+b*c)^2-1/6*d^(5/3)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(2/3)/(-a*d+b*c)^2-1/9*b^(2
/3)*(-5*a*d+2*b*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)/(-a*d+b*c)^2*3^(1/2)-1/3*d^(5/3)*
arctan(1/3*(c^(1/3)-2*d^(1/3)*x)/c^(1/3)*3^(1/2))/c^(2/3)/(-a*d+b*c)^2*3^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {425, 536, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (2 b c-5 a d)}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^3\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^2*(c + d*x^3)),x]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^3)) - (b^(2/3)*(2*b*c - 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)
)])/(3*Sqrt[3]*a^(5/3)*(b*c - a*d)^2) - (d^(5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c
^(2/3)*(b*c - a*d)^2) + (b^(2/3)*(2*b*c - 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*(b*c - a*d)^2) + (d^(5/3
)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*(b*c - a*d)^2) - (b^(2/3)*(2*b*c - 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(18*a^(5/3)*(b*c - a*d)^2) - (d^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^
(2/3)*(b*c - a*d)^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )} \, dx &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}-\frac {\int \frac {-2 b c+3 a d-2 b d x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{3 a (b c-a d)}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {d^2 \int \frac {1}{c+d x^3} \, dx}{(b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{a+b x^3} \, dx}{3 a (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {d^2 \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} (b c-a d)^2}+\frac {d^2 \int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{2/3} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {d^{5/3} \int \frac {-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{2/3} (b c-a d)^2}+\frac {d^2 \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} (b c-a d)^2}-\frac {\left (b^{2/3} (2 b c-5 a d)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{5/3} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac {d^{5/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{2/3} (b c-a d)^2}+\frac {\left (b^{2/3} (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}-\frac {b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 337, normalized size = 0.97 \begin {gather*} \frac {6 a^{2/3} b c^{2/3} (b c-a d) x-2 \sqrt {3} b^{2/3} c^{2/3} (2 b c-5 a d) \left (a+b x^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-6 \sqrt {3} a^{5/3} d^{5/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 b^{2/3} c^{2/3} (2 b c-5 a d) \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-b^{2/3} c^{2/3} (2 b c-5 a d) \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-3 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 a^{5/3} c^{2/3} (b c-a d)^2 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^3)^2*(c + d*x^3)),x]

[Out]

(6*a^(2/3)*b*c^(2/3)*(b*c - a*d)*x - 2*Sqrt[3]*b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]] - 6*Sqrt[3]*a^(5/3)*d^(5/3)*(a + b*x^3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] +
2*b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x] + 6*a^(5/3)*d^(5/3)*(a + b*x^3)*Log[c^
(1/3) + d^(1/3)*x] - b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
] - 3*a^(5/3)*d^(5/3)*(a + b*x^3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(18*a^(5/3)*c^(2/3)*(b*c - a
*d)^2*(a + b*x^3))

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Maple [A]
time = 0.34, size = 247, normalized size = 0.71

method	result	size
default	\(-\frac {b \left (\frac {\left (a d -b c \right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {\left (5 a d -2 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}\right )}{\left (a d -b c \right )^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) d^{2}}{\left (a d -b c \right )^{2}}\)	\(247\)
risch	\(-\frac {b x}{3 a \left (a d -b c \right ) \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (c^{2} d^{6} a^{6}-6 a^{5} b \,c^{3} d^{5}+15 a^{4} b^{2} c^{4} d^{4}-20 a^{3} b^{3} c^{5} d^{3}+15 a^{2} b^{4} c^{6} d^{2}-6 a \,b^{5} c^{7} d +b^{6} c^{8}\right ) \textit {\_Z}^{3}-d^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-27 a^{11} d^{9}+162 a^{10} b c \,d^{8}-360 a^{9} b^{2} c^{2} d^{7}+252 a^{8} b^{3} c^{3} d^{6}+378 a^{7} b^{4} c^{4} d^{5}-1008 a^{6} b^{5} c^{5} d^{4}+1008 a^{5} b^{6} c^{6} d^{3}-540 a^{4} b^{7} c^{7} d^{2}+153 a^{3} b^{8} c^{8} d -18 a^{2} b^{9} c^{9}\right ) \textit {\_R}^{3}-170 a^{3} b^{2} d^{6}+168 a^{2} b^{3} c \,d^{5}-60 a \,b^{4} c^{2} d^{4}+8 b^{5} c^{3} d^{3}\right ) x +\left (-27 a^{13} c \,d^{9}+189 a^{12} b \,c^{2} d^{8}-540 a^{11} b^{2} c^{3} d^{7}+756 a^{10} b^{3} c^{4} d^{6}-378 a^{9} b^{4} c^{5} d^{5}-378 a^{8} b^{5} c^{6} d^{4}+756 a^{7} b^{6} c^{7} d^{3}-540 a^{6} b^{7} c^{8} d^{2}+189 a^{5} b^{8} c^{9} d -27 a^{4} b^{9} c^{10}\right ) \textit {\_R}^{4}+\left (27 a^{6} b \,d^{7}-179 a^{5} b^{2} c \,d^{6}+427 a^{4} b^{3} c^{2} d^{5}-485 a^{3} b^{4} c^{3} d^{4}+278 a^{2} b^{5} c^{4} d^{3}-76 a \,b^{6} c^{5} d^{2}+8 b^{7} c^{6} d \right ) \textit {\_R} \right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (d^{6} a^{11}-6 c \,d^{5} a^{10} b +15 c^{2} d^{4} a^{9} b^{2}-20 c^{3} d^{3} a^{8} b^{3}+15 c^{4} d^{2} a^{7} b^{4}-6 c^{5} d \,a^{6} b^{5}+a^{5} b^{6} c^{6}\right ) \textit {\_Z}^{3}+125 a^{3} b^{2} d^{3}-150 a^{2} b^{3} c \,d^{2}+60 a \,b^{4} c^{2} d -8 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{11} d^{9}+18 a^{10} b c \,d^{8}-40 a^{9} b^{2} c^{2} d^{7}+28 a^{8} b^{3} c^{3} d^{6}+42 a^{7} b^{4} c^{4} d^{5}-112 a^{6} b^{5} c^{5} d^{4}+112 a^{5} b^{6} c^{6} d^{3}-60 a^{4} b^{7} c^{7} d^{2}+17 a^{3} b^{8} c^{8} d -2 a^{2} b^{9} c^{9}\right ) \textit {\_R}^{3}-510 a^{3} b^{2} d^{6}+504 a^{2} b^{3} c \,d^{5}-180 a \,b^{4} c^{2} d^{4}+24 b^{5} c^{3} d^{3}\right ) x +\left (-a^{13} c \,d^{9}+7 a^{12} b \,c^{2} d^{8}-20 a^{11} b^{2} c^{3} d^{7}+28 a^{10} b^{3} c^{4} d^{6}-14 a^{9} b^{4} c^{5} d^{5}-14 a^{8} b^{5} c^{6} d^{4}+28 a^{7} b^{6} c^{7} d^{3}-20 a^{6} b^{7} c^{8} d^{2}+7 a^{5} b^{8} c^{9} d -a^{4} b^{9} c^{10}\right ) \textit {\_R}^{4}+\left (27 a^{6} b \,d^{7}-179 a^{5} b^{2} c \,d^{6}+427 a^{4} b^{3} c^{2} d^{5}-485 a^{3} b^{4} c^{3} d^{4}+278 a^{2} b^{5} c^{4} d^{3}-76 a \,b^{6} c^{5} d^{2}+8 b^{7} c^{6} d \right ) \textit {\_R} \right )\right )}{9}\)	\(1066\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^2/(d*x^3+c),x,method=_RETURNVERBOSE)

[Out]

-b/(a*d-b*c)^2*(1/3*(a*d-b*c)/a*x/(b*x^3+a)+1/3*(5*a*d-2*b*c)/a*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/
b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))
+(1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))-1/6/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))+1/3/d/(c/d)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1)))*d^2/(a*d-b*c)^2

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Maxima [A]
time = 0.51, size = 489, normalized size = 1.41 \begin {gather*} \frac {\sqrt {3} {\left (2 \, b c - 5 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {b x}{3 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{3}\right )}} - \frac {{\left (2 \, b c - 5 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (2 \, b c - 5 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^2/(d*x^3+c),x, algorithm="maxima")

[Out]

1/9*sqrt(3)*(2*b*c - 5*a*d)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((a*b^2*c^2*(a/b)^(1/3) - 2*a^
2*b*c*d*(a/b)^(1/3) + a^3*d^2*(a/b)^(1/3))*(a/b)^(1/3)) + 1/3*sqrt(3)*d*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/3))
/(c/d)^(1/3))/((b^2*c^2*(c/d)^(1/3) - 2*a*b*c*d*(c/d)^(1/3) + a^2*d^2*(c/d)^(1/3))*(c/d)^(1/3)) + 1/3*b*x/(a^2
*b*c - a^3*d + (a*b^2*c - a^2*b*d)*x^3) - 1/18*(2*b*c - 5*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2*c
^2*(a/b)^(2/3) - 2*a^2*b*c*d*(a/b)^(2/3) + a^3*d^2*(a/b)^(2/3)) - 1/6*d*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))
/(b^2*c^2*(c/d)^(2/3) - 2*a*b*c*d*(c/d)^(2/3) + a^2*d^2*(c/d)^(2/3)) + 1/9*(2*b*c - 5*a*d)*log(x + (a/b)^(1/3)
)/(a*b^2*c^2*(a/b)^(2/3) - 2*a^2*b*c*d*(a/b)^(2/3) + a^3*d^2*(a/b)^(2/3)) + 1/3*d*log(x + (c/d)^(1/3))/(b^2*c^
2*(c/d)^(2/3) - 2*a*b*c*d*(c/d)^(2/3) + a^2*d^2*(c/d)^(2/3))

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Fricas [A]
time = 22.04, size = 440, normalized size = 1.27 \begin {gather*} -\frac {2 \, \sqrt {3} {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 6 \, \sqrt {3} {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b^{2} c - a b d\right )} x}{18 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^2/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/18*(2*sqrt(3)*((2*b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 5*a^2*d)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^
2/a^2)^(2/3) - sqrt(3)*b)/b) - 6*sqrt(3)*(a*b*d*x^3 + a^2*d)*(d^2/c^2)^(1/3)*arctan(1/3*(2*sqrt(3)*c*x*(d^2/c^
2)^(2/3) - sqrt(3)*d)/d) - ((2*b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 5*a^2*d)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*
(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) + 3*(a*b*d*x^3 + a^2*d)*(d^2/c^2)^(1/3)*log(d^2*x^2 - c*d*x*(d^2/c^2)
^(1/3) + c^2*(d^2/c^2)^(2/3)) + 2*((2*b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 5*a^2*d)*(-b^2/a^2)^(1/3)*log(b*x - a*(
-b^2/a^2)^(1/3)) - 6*(a*b*d*x^3 + a^2*d)*(d^2/c^2)^(1/3)*log(d*x + c*(d^2/c^2)^(1/3)) - 6*(b^2*c - a*b*d)*x)/(
a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a)**2/(d*x**3+c),x)

[Out]

Timed out

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Giac [A]
time = 0.88, size = 443, normalized size = 1.28 \begin {gather*} -\frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c^{3} - 2 \, \sqrt {3} a b c^{2} d + \sqrt {3} a^{2} c d^{2}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} - \frac {{\left (2 \, b^{2} c - 5 \, a b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (\sqrt {3} a^{2} b^{2} c^{2} - 2 \, \sqrt {3} a^{3} b c d + \sqrt {3} a^{4} d^{2}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac {b x}{3 \, {\left (b x^{3} + a\right )} {\left (a b c - a^{2} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^2/(d*x^3+c),x, algorithm="giac")

[Out]

-1/3*d^2*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2) + (-c*d^2)^(1/3)*d*arctan
(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b^2*c^3 - 2*sqrt(3)*a*b*c^2*d + sqrt(3)*a^2*c*d^2) +
1/6*(-c*d^2)^(1/3)*d*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2) - 1/9*(2*b^2
*c - 5*a*b*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2) + 1/3*(2*(-a*b^2)^
(1/3)*b*c - 5*(-a*b^2)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^2*b^2*c^2 -
2*sqrt(3)*a^3*b*c*d + sqrt(3)*a^4*d^2) + 1/18*(2*(-a*b^2)^(1/3)*b*c - 5*(-a*b^2)^(1/3)*a*d)*log(x^2 + x*(-a/b
)^(1/3) + (-a/b)^(2/3))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2) + 1/3*b*x/((b*x^3 + a)*(a*b*c - a^2*d))

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Mupad [B]
time = 15.93, size = 2492, normalized size = 7.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^3)^2*(c + d*x^3)),x)

[Out]

log((((((27*b^3*d^3*x*(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d))/a + 27*a*b^3*c*d^3*(a*d + b*c)*(a*d -
b*c)^4*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(
2/3))/81 - (b^4*d^4*(27*a^3*d^3 - 8*b^3*c^3 + 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c))*(-(b^2*
(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))/9 + (2*b^5*d^6*x*(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*c^2*d - 84*
a^2*b*c*d^2))/(9*a^3*(a*d - b*c)^4))*((8*b^5*c^3 - 125*a^3*b^2*d^3 + 150*a^2*b^3*c*d^2 - 60*a*b^4*c^2*d)/(729*
a^11*d^6 + 729*a^5*b^6*c^6 - 4374*a^6*b^5*c^5*d + 10935*a^7*b^4*c^4*d^2 - 14580*a^8*b^3*c^3*d^3 + 10935*a^9*b^
2*c^2*d^4 - 4374*a^10*b*c*d^5))^(1/3) + log((((((27*b^3*d^3*x*(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d
))/a + 81*a*b^3*c*d^3*(a*d + b*c)*(a*d - b*c)^4*(d^5/(c^2*(a*d - b*c)^6))^(1/3))*(d^5/(c^2*(a*d - b*c)^6))^(2/
3))/9 - (b^4*d^4*(27*a^3*d^3 - 8*b^3*c^3 + 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c))*(d^5/(c^2*
(a*d - b*c)^6))^(1/3))/3 + (2*b^5*d^6*x*(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*c^2*d - 84*a^2*b*c*d^2))/(9*a^3*(a*
d - b*c)^4))*(d^5/(27*b^6*c^8 + 27*a^6*c^2*d^6 - 162*a^5*b*c^3*d^5 + 405*a^2*b^4*c^6*d^2 - 540*a^3*b^3*c^5*d^3
+ 405*a^4*b^2*c^4*d^4 - 162*a*b^5*c^7*d))^(1/3) + (log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i - 1)^2*((27*b^3*d^3*x*
(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d))/a + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c
)^4*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))/2)*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(2
/3))/324 - (b^4*d^4*(27*a^3*d^3 - 8*b^3*c^3 + 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c))*(-(b^2*
(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))/18 + (2*b^5*d^6*x*(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*c^2*d - 84
*a^2*b*c*d^2))/(9*a^3*(a*d - b*c)^4))*(3^(1/2)*1i - 1)*((8*b^5*c^3 - 125*a^3*b^2*d^3 + 150*a^2*b^3*c*d^2 - 60*
a*b^4*c^2*d)/(729*a^11*d^6 + 729*a^5*b^6*c^6 - 4374*a^6*b^5*c^5*d + 10935*a^7*b^4*c^4*d^2 - 14580*a^8*b^3*c^3*
d^3 + 10935*a^9*b^2*c^2*d^4 - 4374*a^10*b*c*d^5))^(1/3))/2 - (log(((3^(1/2)*1i + 1)*(((3^(1/2)*1i + 1)^2*((27*
b^3*d^3*x*(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d))/a - (27*a*b^3*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*
(a*d - b*c)^4*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))/2)*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b
*c)^6))^(2/3))/324 - (b^4*d^4*(27*a^3*d^3 - 8*b^3*c^3 + 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c
))*(-(b^2*(5*a*d - 2*b*c)^3)/(a^5*(a*d - b*c)^6))^(1/3))/18 - (2*b^5*d^6*x*(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*
c^2*d - 84*a^2*b*c*d^2))/(9*a^3*(a*d - b*c)^4))*(3^(1/2)*1i + 1)*((8*b^5*c^3 - 125*a^3*b^2*d^3 + 150*a^2*b^3*c
*d^2 - 60*a*b^4*c^2*d)/(729*a^11*d^6 + 729*a^5*b^6*c^6 - 4374*a^6*b^5*c^5*d + 10935*a^7*b^4*c^4*d^2 - 14580*a^
8*b^3*c^3*d^3 + 10935*a^9*b^2*c^2*d^4 - 4374*a^10*b*c*d^5))^(1/3))/2 + (log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i -
1)^2*((27*b^3*d^3*x*(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d))/a + (81*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a
*d + b*c)*(a*d - b*c)^4*(d^5/(c^2*(a*d - b*c)^6))^(1/3))/2)*(d^5/(c^2*(a*d - b*c)^6))^(2/3))/36 - (b^4*d^4*(27
*a^3*d^3 - 8*b^3*c^3 + 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c))*(d^5/(c^2*(a*d - b*c)^6))^(1/3
))/6 + (2*b^5*d^6*x*(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*c^2*d - 84*a^2*b*c*d^2))/(9*a^3*(a*d - b*c)^4))*(3^(1/2
)*1i - 1)*(d^5/(27*b^6*c^8 + 27*a^6*c^2*d^6 - 162*a^5*b*c^3*d^5 + 405*a^2*b^4*c^6*d^2 - 540*a^3*b^3*c^5*d^3 +
405*a^4*b^2*c^4*d^4 - 162*a*b^5*c^7*d))^(1/3))/2 - (log(((3^(1/2)*1i + 1)*(((3^(1/2)*1i + 1)^2*((27*b^3*d^3*x*
(a*d - b*c)^3*(3*a^2*d^2 - 2*b^2*c^2 + 3*a*b*c*d))/a - (81*a*b^3*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c
)^4*(d^5/(c^2*(a*d - b*c)^6))^(1/3))/2)*(d^5/(c^2*(a*d - b*c)^6))^(2/3))/36 - (b^4*d^4*(27*a^3*d^3 - 8*b^3*c^3
+ 52*a*b^2*c^2*d - 98*a^2*b*c*d^2))/(3*a^4*d - 3*a^3*b*c))*(d^5/(c^2*(a*d - b*c)^6))^(1/3))/6 - (2*b^5*d^6*x*
(85*a^3*d^3 - 4*b^3*c^3 + 30*a*b^2*c^2*d - 84*a^2*b*c*d^2))/(9*a^3*(a*d - b*c)^4))*(3^(1/2)*1i + 1)*(d^5/(27*b
^6*c^8 + 27*a^6*c^2*d^6 - 162*a^5*b*c^3*d^5 + 405*a^2*b^4*c^6*d^2 - 540*a^3*b^3*c^5*d^3 + 405*a^4*b^2*c^4*d^4
- 162*a*b^5*c^7*d))^(1/3))/2 - (b*x)/(3*a*(a + b*x^3)*(a*d - b*c))

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