Optimal. Leaf size=46 \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]
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Rubi [A] time = 0.0315299, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ -\frac{b^2}{3 a^3 \left (a x^3+b\right )}-\frac{2 b \log \left (a x^3+b\right )}{3 a^3}+\frac{x^3}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x^3}\right )^2} \, dx &=\int \frac{x^8}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{a^2}+\frac{b^2}{a^2 (b+a x)^2}-\frac{2 b}{a^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 a^2}-\frac{b^2}{3 a^3 \left (b+a x^3\right )}-\frac{2 b \log \left (b+a x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0152276, size = 38, normalized size = 0.83 \[ \frac{-\frac{b^2}{a x^3+b}-2 b \log \left (a x^3+b\right )+a x^3}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3} \left ( a{x}^{3}+b \right ) }}-{\frac{2\,b\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98249, size = 58, normalized size = 1.26 \begin{align*} -\frac{b^{2}}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} + \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b \log \left (a x^{3} + b\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43891, size = 113, normalized size = 2.46 \begin{align*} \frac{a^{2} x^{6} + a b x^{3} - b^{2} - 2 \,{\left (a b x^{3} + b^{2}\right )} \log \left (a x^{3} + b\right )}{3 \,{\left (a^{4} x^{3} + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.453041, size = 42, normalized size = 0.91 \begin{align*} - \frac{b^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \frac{x^{3}}{3 a^{2}} - \frac{2 b \log{\left (a x^{3} + b \right )}}{3 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14552, size = 55, normalized size = 1.2 \begin{align*} \frac{x^{3}}{3 \, a^{2}} - \frac{2 \, b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{3}} - \frac{b^{2}}{3 \,{\left (a x^{3} + b\right )} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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