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

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

[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)*A
rcTan[(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)

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Rubi [A]  time = 0.589269, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}+\frac{b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}-\frac{b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} (b c-a d)^2}-\frac{d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac{d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac{d^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} (b c-a d)^2}+\frac{b x}{3 a \left (a+b x^3\right ) (b c-a d)} \]

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)*A
rcTan[(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)

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Rubi in Sympy [A]  time = 109.102, size = 321, normalized size = 0.93 \[ \frac{d^{\frac{5}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )^{2}} - \frac{d^{\frac{5}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} \left (a d - b c\right )^{2}} - \frac{\sqrt{3} d^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )^{2}} - \frac{b x}{3 a \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{b^{\frac{2}{3}} \left (5 a d - 2 b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{2}{3}} \left (5 a d - 2 b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} \left (a d - b c\right )^{2}} + \frac{\sqrt{3} b^{\frac{2}{3}} \left (5 a d - 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(5/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(2/3)*(a*d - b*c)**2) - d**(5/3)*log(c
**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(2/3)*(a*d - b*c)**2) - sqr
t(3)*d**(5/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*(
a*d - b*c)**2) - b*x/(3*a*(a + b*x**3)*(a*d - b*c)) - b**(2/3)*(5*a*d - 2*b*c)*l
og(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*(a*d - b*c)**2) + b**(2/3)*(5*a*d - 2*b*c)
*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(5/3)*(a*d - b*c)**2
) + sqrt(3)*b**(2/3)*(5*a*d - 2*b*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/
a**(1/3))/(9*a**(5/3)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.360038, size = 337, normalized size = 0.97 \[ \frac{-b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-3 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )+6 a^{2/3} b c^{2/3} x (b c-a d)+6 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-6 \sqrt{3} a^{5/3} d^{5/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )+2 b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{5/3} c^{2/3} \left (a+b x^3\right ) (b c-a d)^2} \]

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 = 0.017, size = 406, normalized size = 1.2 \[ -{\frac{bxd}{3\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}xc}{3\, \left ( ad-bc \right ) ^{2}a \left ( b{x}^{3}+a \right ) }}-{\frac{5\,d}{9\, \left ( ad-bc \right ) ^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,bc}{9\, \left ( ad-bc \right ) ^{2}a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,d}{18\, \left ( ad-bc \right ) ^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{bc}{9\, \left ( ad-bc \right ) ^{2}a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,\sqrt{3}d}{9\, \left ( ad-bc \right ) ^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,b\sqrt{3}c}{9\, \left ( ad-bc \right ) ^{2}a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d}{3\, \left ( ad-bc \right ) ^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{6\, \left ( ad-bc \right ) ^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}d}{3\, \left ( ad-bc \right ) ^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)^2/(d*x^3+c),x)

[Out]

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

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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)^2*(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.40408, size = 633, normalized size = 1.83 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 3 \, \sqrt{3}{\left (a b d x^{3} + a^{2} d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, \sqrt{3}{\left (a b d x^{3} + a^{2} d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, \sqrt{3}{\left (b^{2} c - a b d\right )} x + 6 \,{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 18 \,{\left (a b d x^{3} + a^{2} d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right )\right )}}{54 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )}} \]

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]

1/54*sqrt(3)*(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)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 3*sqrt(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*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)*log(b*x - a*(-b^2/a^2)^(1/3)) + 6*sqrt(3)*(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*sqrt(3)*(b^2*c - a*b*d)*x + 6*((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)*b*x +
sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) - 18*(a*b*d*x^3 + a^2*d)*(d^2/
c^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(d^2/c^2)^(1/3))/(c*(d^2/c^2)^
(1/3))))/(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)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.228681, size = 598, normalized size = 1.73 \[ -\frac{d^{2} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c^{3} - 2 \, \sqrt{3} a b c^{2} d + \sqrt{3} a^{2} c d^{2}} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} - \frac{{\left (2 \, b^{2} c - 5 \, a b d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \,{\left (\sqrt{3} a^{2} b^{2} c^{2} - 2 \, \sqrt{3} a^{3} b c d + \sqrt{3} a^{4} d^{2}\right )}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac{b x}{3 \,{\left (b x^{3} + a\right )}{\left (a b c - a^{2} d\right )}} \]

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)*ln(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))/(sq
rt(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*
ln(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)*ln(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)*
ln(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))