3.117 \(\int \frac{1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=299 \[ -\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{4/3} (b c-a d)}-\frac{1}{a c x} \]

[Out]

-(1/(a*c*x)) + (b^(4/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt
[3]*a^(4/3)*(b*c - a*d)) - (d^(4/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1
/3))])/(Sqrt[3]*c^(4/3)*(b*c - a*d)) + (b^(4/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(
4/3)*(b*c - a*d)) - (d^(4/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(4/3)*(b*c - a*d)) -
(b^(4/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*(b*c - a*d)
) + (d^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(4/3)*(b*c - a
*d))

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Rubi [A]  time = 0.691927, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{4/3} (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*c*x)) + (b^(4/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt
[3]*a^(4/3)*(b*c - a*d)) - (d^(4/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1
/3))])/(Sqrt[3]*c^(4/3)*(b*c - a*d)) + (b^(4/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(
4/3)*(b*c - a*d)) - (d^(4/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(4/3)*(b*c - a*d)) -
(b^(4/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*(b*c - a*d)
) + (d^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(4/3)*(b*c - a
*d))

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Rubi in Sympy [A]  time = 115.728, size = 267, normalized size = 0.89 \[ \frac{d^{\frac{4}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{4}{3}} \left (a d - b c\right )} - \frac{d^{\frac{4}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{4}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} d^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{4}{3}} \left (a d - b c\right )} - \frac{1}{a c x} - \frac{b^{\frac{4}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{4}{3}} \left (a d - b c\right )} + \frac{b^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{4}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(4/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(4/3)*(a*d - b*c)) - d**(4/3)*log(c**(
2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(4/3)*(a*d - b*c)) + sqrt(3)*d
**(4/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(4/3)*(a*d -
b*c)) - 1/(a*c*x) - b**(4/3)*log(a**(1/3) + b**(1/3)*x)/(3*a**(4/3)*(a*d - b*c))
+ b**(4/3)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(4/3)*(a*d
- b*c)) - sqrt(3)*b**(4/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))
/(3*a**(4/3)*(a*d - b*c))

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Mathematica [A]  time = 0.309663, size = 244, normalized size = 0.82 \[ \frac{\frac{b^{4/3} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 b^{4/3} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} b^{4/3} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}+\frac{6 b}{a}-\frac{d^{4/3} x \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{4/3}}+\frac{2 d^{4/3} x \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{4/3}}+\frac{2 \sqrt{3} d^{4/3} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{4/3}}-\frac{6 d}{c}}{6 a d x-6 b c x} \]

Antiderivative was successfully verified.

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

[Out]

((6*b)/a - (6*d)/c - (2*Sqrt[3]*b^(4/3)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr
t[3]])/a^(4/3) + (2*Sqrt[3]*d^(4/3)*x*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]
])/c^(4/3) - (2*b^(4/3)*x*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (2*d^(4/3)*x*Log[c
^(1/3) + d^(1/3)*x])/c^(4/3) + (b^(4/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/a^(4/3) - (d^(4/3)*x*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c
^(4/3))/(-6*b*c*x + 6*a*d*x)

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Maple [A]  time = 0.013, size = 257, normalized size = 0.9 \[ -{\frac{b}{3\,a \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b}{6\,a \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}}{3\,a \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{3\,c \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{d}{6\,c \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{3}}{3\,c \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{1}{acx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252817, size = 377, normalized size = 1.26 \[ \frac{\sqrt{3}{\left (\sqrt{3} b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + \sqrt{3} a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (\frac{d}{c}\right )^{\frac{2}{3}} + c \left (\frac{d}{c}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d}{c}\right )^{\frac{2}{3}}\right ) - 6 \, b c x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 6 \, a d x \left (\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d}{c}\right )^{\frac{2}{3}}}{3 \, c \left (\frac{d}{c}\right )^{\frac{2}{3}}}\right ) - 6 \, \sqrt{3}{\left (b c - a d\right )}\right )}}{18 \,{\left (a b c^{2} - a^{2} c d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 108.363, size = 661, normalized size = 2.21 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{7} d^{3} - 81 a^{6} b c d^{2} + 81 a^{5} b^{2} c^{2} d - 27 a^{4} b^{3} c^{3}\right ) + b^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{12} c^{4} d^{8} + 1215 t^{5} a^{11} b c^{5} d^{7} - 2430 t^{5} a^{10} b^{2} c^{6} d^{6} + 2673 t^{5} a^{9} b^{3} c^{7} d^{5} - 2430 t^{5} a^{8} b^{4} c^{8} d^{4} + 2673 t^{5} a^{7} b^{5} c^{9} d^{3} - 2430 t^{5} a^{6} b^{6} c^{10} d^{2} + 1215 t^{5} a^{5} b^{7} c^{11} d - 243 t^{5} a^{4} b^{8} c^{12} + 9 t^{2} a^{9} d^{9} - 18 t^{2} a^{8} b c d^{8} + 9 t^{2} a^{7} b^{2} c^{2} d^{7} + 9 t^{2} a^{2} b^{7} c^{7} d^{2} - 18 t^{2} a b^{8} c^{8} d + 9 t^{2} b^{9} c^{9}}{a^{4} b^{3} d^{7} + b^{7} c^{4} d^{3}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c^{4} d^{3} - 81 a^{2} b c^{5} d^{2} + 81 a b^{2} c^{6} d - 27 b^{3} c^{7}\right ) - d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{12} c^{4} d^{8} + 1215 t^{5} a^{11} b c^{5} d^{7} - 2430 t^{5} a^{10} b^{2} c^{6} d^{6} + 2673 t^{5} a^{9} b^{3} c^{7} d^{5} - 2430 t^{5} a^{8} b^{4} c^{8} d^{4} + 2673 t^{5} a^{7} b^{5} c^{9} d^{3} - 2430 t^{5} a^{6} b^{6} c^{10} d^{2} + 1215 t^{5} a^{5} b^{7} c^{11} d - 243 t^{5} a^{4} b^{8} c^{12} + 9 t^{2} a^{9} d^{9} - 18 t^{2} a^{8} b c d^{8} + 9 t^{2} a^{7} b^{2} c^{2} d^{7} + 9 t^{2} a^{2} b^{7} c^{7} d^{2} - 18 t^{2} a b^{8} c^{8} d + 9 t^{2} b^{9} c^{9}}{a^{4} b^{3} d^{7} + b^{7} c^{4} d^{3}} \right )} \right )\right )} - \frac{1}{a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**3*(27*a**7*d**3 - 81*a**6*b*c*d**2 + 81*a**5*b**2*c**2*d - 27*a**4*b
**3*c**3) + b**4, Lambda(_t, _t*log(x + (-243*_t**5*a**12*c**4*d**8 + 1215*_t**5
*a**11*b*c**5*d**7 - 2430*_t**5*a**10*b**2*c**6*d**6 + 2673*_t**5*a**9*b**3*c**7
*d**5 - 2430*_t**5*a**8*b**4*c**8*d**4 + 2673*_t**5*a**7*b**5*c**9*d**3 - 2430*_
t**5*a**6*b**6*c**10*d**2 + 1215*_t**5*a**5*b**7*c**11*d - 243*_t**5*a**4*b**8*c
**12 + 9*_t**2*a**9*d**9 - 18*_t**2*a**8*b*c*d**8 + 9*_t**2*a**7*b**2*c**2*d**7
+ 9*_t**2*a**2*b**7*c**7*d**2 - 18*_t**2*a*b**8*c**8*d + 9*_t**2*b**9*c**9)/(a**
4*b**3*d**7 + b**7*c**4*d**3)))) + RootSum(_t**3*(27*a**3*c**4*d**3 - 81*a**2*b*
c**5*d**2 + 81*a*b**2*c**6*d - 27*b**3*c**7) - d**4, Lambda(_t, _t*log(x + (-243
*_t**5*a**12*c**4*d**8 + 1215*_t**5*a**11*b*c**5*d**7 - 2430*_t**5*a**10*b**2*c*
*6*d**6 + 2673*_t**5*a**9*b**3*c**7*d**5 - 2430*_t**5*a**8*b**4*c**8*d**4 + 2673
*_t**5*a**7*b**5*c**9*d**3 - 2430*_t**5*a**6*b**6*c**10*d**2 + 1215*_t**5*a**5*b
**7*c**11*d - 243*_t**5*a**4*b**8*c**12 + 9*_t**2*a**9*d**9 - 18*_t**2*a**8*b*c*
d**8 + 9*_t**2*a**7*b**2*c**2*d**7 + 9*_t**2*a**2*b**7*c**7*d**2 - 18*_t**2*a*b*
*8*c**8*d + 9*_t**2*b**9*c**9)/(a**4*b**3*d**7 + b**7*c**4*d**3)))) - 1/(a*c*x)

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GIAC/XCAS [A]  time = 0.228523, size = 412, normalized size = 1.38 \[ \frac{b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{3} - a c^{2} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{2} b c - \sqrt{3} a^{3} d} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{3} - \sqrt{3} a c^{2} d} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{2} b c - a^{3} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{1}{a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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