Optimal. Leaf size=52 \[ -\frac{a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac{2 a}{3 b^3 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^3} \]
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Rubi [A] time = 0.0373049, antiderivative size = 52, 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} \[ -\frac{a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac{2 a}{3 b^3 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^3} \]
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 )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^3}-\frac{2 a}{b^2 (a+b x)^2}+\frac{1}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac{2 a}{3 b^3 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0151773, size = 39, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 b x^3\right )}{\left (a+b x^3\right )^2}+2 \log \left (a+b x^3\right )}{6 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,a}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{\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 = 0.957314, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \, a b x^{3} + 3 \, a^{2}}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{\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 = 1.47579, size = 143, normalized size = 2.75 \begin{align*} \frac{4 \, a b x^{3} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.83584, size = 53, normalized size = 1.02 \begin{align*} \frac{3 a^{2} + 4 a b x^{3}}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} + \frac{\log{\left (a + b x^{3} \right )}}{3 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17542, size = 57, normalized size = 1.1 \begin{align*} \frac{\log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{3 \, b x^{6} + 2 \, a x^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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