3.1987 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^2 x^6} \, dx\)

Optimal. Leaf size=134 \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}+\frac{x}{3 b \left (a x^3+b\right )} \]

[Out]

x/(3*b*(b + a*x^3)) - (2*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*S
qrt[3]*a^(1/3)*b^(5/3)) + (2*Log[b^(1/3) + a^(1/3)*x])/(9*a^(1/3)*b^(5/3)) - Log
[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(9*a^(1/3)*b^(5/3))

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Rubi [A]  time = 0.160351, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}+\frac{x}{3 b \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)^2*x^6),x]

[Out]

x/(3*b*(b + a*x^3)) - (2*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*S
qrt[3]*a^(1/3)*b^(5/3)) + (2*Log[b^(1/3) + a^(1/3)*x])/(9*a^(1/3)*b^(5/3)) - Log
[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(9*a^(1/3)*b^(5/3))

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Rubi in Sympy [A]  time = 28.6642, size = 124, normalized size = 0.93 \[ \frac{x}{3 b \left (a x^{3} + b\right )} + \frac{2 \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} - \frac{\log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)**2/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.117481, size = 118, normalized size = 0.88 \[ \frac{-\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{\sqrt [3]{a}}+\frac{3 b^{2/3} x}{a x^3+b}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)^2*x^6),x]

[Out]

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

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Maple [A]  time = 0.007, size = 115, normalized size = 0.9 \[{\frac{x}{3\,b \left ( a{x}^{3}+b \right ) }}+{\frac{2}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^2/x^6,x)

[Out]

1/3*x/b/(a*x^3+b)+2/9/b/a/(b/a)^(2/3)*ln(x+(b/a)^(1/3))-1/9/b/a/(b/a)^(2/3)*ln(x
^2-x*(b/a)^(1/3)+(b/a)^(2/3))+2/9/b/a/(b/a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/
(b/a)^(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/((a + b/x^3)^2*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238254, size = 182, normalized size = 1.36 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (a x^{3} + b\right )} \log \left (\left (a b^{2}\right )^{\frac{2}{3}} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} b x + b^{2}\right ) - 2 \, \sqrt{3}{\left (a x^{3} + b\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} x + b\right ) - 6 \,{\left (a x^{3} + b\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}} x - \sqrt{3} b}{3 \, b}\right ) - 3 \, \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}} x\right )}}{27 \,{\left (a b x^{3} + b^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^2*x^6),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(sqrt(3)*(a*x^3 + b)*log((a*b^2)^(2/3)*x^2 - (a*b^2)^(1/3)*b*x + b
^2) - 2*sqrt(3)*(a*x^3 + b)*log((a*b^2)^(1/3)*x + b) - 6*(a*x^3 + b)*arctan(1/3*
(2*sqrt(3)*(a*b^2)^(1/3)*x - sqrt(3)*b)/b) - 3*sqrt(3)*(a*b^2)^(1/3)*x)/((a*b*x^
3 + b^2)*(a*b^2)^(1/3))

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Sympy [A]  time = 1.64388, size = 39, normalized size = 0.29 \[ \frac{x}{3 a b x^{3} + 3 b^{2}} + \operatorname{RootSum}{\left (729 t^{3} a b^{5} - 8, \left ( t \mapsto t \log{\left (\frac{9 t b^{2}}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)**2/x**6,x)

[Out]

x/(3*a*b*x**3 + 3*b**2) + RootSum(729*_t**3*a*b**5 - 8, Lambda(_t, _t*log(9*_t*b
**2/2 + x)))

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GIAC/XCAS [A]  time = 0.233769, size = 171, normalized size = 1.28 \[ -\frac{2 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, b^{2}} + \frac{x}{3 \,{\left (a x^{3} + b\right )} b} + \frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{2}} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{9 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^2*x^6),x, algorithm="giac")

[Out]

-2/9*(-b/a)^(1/3)*ln(abs(x - (-b/a)^(1/3)))/b^2 + 1/3*x/((a*x^3 + b)*b) + 2/9*sq
rt(3)*(-a^2*b)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/(a*b^
2) + 1/9*(-a^2*b)^(1/3)*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/(a*b^2)