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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.613004, 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 \[ -\frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)^2}+\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)^2}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} (b c-a d)^2}+\frac{d^{2/3} (5 b c-2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} (b c-a d)^2}-\frac{d^{2/3} (5 b c-2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^2}+\frac{d^{2/3} (5 b c-2 a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^2}-\frac{d x}{3 c \left (c+d x^3\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 108.441, size = 321, normalized size = 0.93 \[ \frac{d x}{3 c \left (c + d x^{3}\right ) \left (a d - b c\right )} + \frac{d^{\frac{2}{3}} \left (2 a d - 5 b c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{9 c^{\frac{5}{3}} \left (a d - b c\right )^{2}} - \frac{d^{\frac{2}{3}} \left (2 a d - 5 b c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{18 c^{\frac{5}{3}} \left (a d - b c\right )^{2}} - \frac{\sqrt{3} d^{\frac{2}{3}} \left (2 a d - 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{9 c^{\frac{5}{3}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{5}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )^{2}} - \frac{b^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} \left (a d - b c\right )^{2}} - \frac{\sqrt{3} b^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.351284, size = 336, normalized size = 0.97 \[ \frac{-3 b^{5/3} c^{5/3} \left (c+d x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} d^{2/3} \left (c+d x^3\right ) (5 b c-2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )+6 a^{2/3} c^{2/3} d x (a d-b c)+2 a^{2/3} d^{2/3} \left (c+d x^3\right ) (2 a d-5 b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} a^{2/3} d^{2/3} \left (c+d x^3\right ) (2 a d-5 b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )+6 b^{5/3} c^{5/3} \left (c+d x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 \sqrt{3} b^{5/3} c^{5/3} \left (c+d x^3\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{2/3} c^{5/3} \left (c+d x^3\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 406, normalized size = 1.2 \[{\frac{b}{3\, \left ( ad-bc \right ) ^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{6\, \left ( ad-bc \right ) ^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{3\, \left ( ad-bc \right ) ^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{d}^{2}xa}{3\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{3}+c \right ) }}-{\frac{dxb}{3\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{3}+c \right ) }}+{\frac{2\,ad}{9\, \left ( ad-bc \right ) ^{2}c}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,b}{9\, \left ( ad-bc \right ) ^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{ad}{9\, \left ( ad-bc \right ) ^{2}c}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,b}{18\, \left ( ad-bc \right ) ^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,d\sqrt{3}a}{9\, \left ( ad-bc \right ) ^{2}c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,b\sqrt{3}}{9\, \left ( ad-bc \right ) ^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)/(d*x^3+c)^2,x)

[Out]

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

_______________________________________________________________________________________

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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 7.36613, size = 625, normalized size = 1.81 \[ -\frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (b c d x^{3} + b c^{2}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - \sqrt{3}{\left ({\left (5 \, b c d - 2 \, a d^{2}\right )} x^{3} + 5 \, b c^{2} - 2 \, a c d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 6 \, \sqrt{3}{\left (b c d x^{3} + b c^{2}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3}{\left ({\left (5 \, b c d - 2 \, a d^{2}\right )} x^{3} + 5 \, b c^{2} - 2 \, a c d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, \sqrt{3}{\left (b c d - a d^{2}\right )} x + 18 \,{\left (b c d x^{3} + b c^{2}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 6 \,{\left ({\left (5 \, b c d - 2 \, a d^{2}\right )} x^{3} + 5 \, b c^{2} - 2 \, a c d\right )} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right )\right )}}{54 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(3*sqrt(3)*(b*c*d*x^3 + b*c^2)*(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)) - sqrt(3)*((5*b*c*d - 2*a*d^2)*x^3 + 5*b
*c^2 - 2*a*c*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)) - 6*sqrt(3)*(b*c*d*x^3 + b*c^2)*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)
^(1/3)) + 2*sqrt(3)*((5*b*c*d - 2*a*d^2)*x^3 + 5*b*c^2 - 2*a*c*d)*(d^2/c^2)^(1/3
)*log(d*x + c*(d^2/c^2)^(1/3)) + 6*sqrt(3)*(b*c*d - a*d^2)*x + 18*(b*c*d*x^3 + b
*c^2)*(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))) - 6*((5*b*c*d - 2*a*d^2)*x^3 + 5*b*c^2 - 2*a*c*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))
))/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3
)*x^3)

_______________________________________________________________________________________

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)/(d*x**3+c)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227324, size = 598, normalized size = 1.73 \[ -\frac{b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b^{2} c^{2} - 2 \, \sqrt{3} a^{2} b c d + \sqrt{3} a^{3} d^{2}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{{\left (5 \, b c d - 2 \, a d^{2}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{9 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac{{\left (5 \, \left (-c d^{2}\right )^{\frac{1}{3}} b c - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \,{\left (\sqrt{3} b^{2} c^{4} - 2 \, \sqrt{3} a b c^{3} d + \sqrt{3} a^{2} c^{2} d^{2}\right )}} - \frac{{\left (5 \, \left (-c d^{2}\right )^{\frac{1}{3}} b c - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{18 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac{d x}{3 \,{\left (d x^{3} + c\right )}{\left (b c^{2} - a c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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