Optimal. Leaf size=46 \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]
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Rubi [A] time = 0.0303931, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b^2}-\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0180864, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+b x^3}-2 a \log \left (a+b x^3\right )+b x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986291, size = 58, normalized size = 1.26 \begin{align*} -\frac{a^{2}}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} + \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a \log \left (b x^{3} + a\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43615, size = 113, normalized size = 2.46 \begin{align*} \frac{b^{2} x^{6} + a b x^{3} - a^{2} - 2 \,{\left (a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.58039, size = 42, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{3 a b^{3} + 3 b^{4} x^{3}} - \frac{2 a \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{x^{3}}{3 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08214, size = 66, normalized size = 1.43 \begin{align*} \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{2 \, a b x^{3} + a^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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