Optimal. Leaf size=61 \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]
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Rubi [A] time = 0.048149, antiderivative size = 61, 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} \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b^3}+\frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0550541, size = 48, normalized size = 0.79 \[ -\frac{\frac{a^2 \left (5 a+6 b x^3\right )}{\left (a+b x^3\right )^2}+6 a \log \left (a+b x^3\right )-2 b x^3}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 58, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{a\ln \left ( b{x}^{3}+a \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973354, size = 89, normalized size = 1.46 \begin{align*} -\frac{6 \, a^{2} b x^{3} + 5 \, a^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac{x^{3}}{3 \, b^{3}} - \frac{a \log \left (b x^{3} + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45777, size = 186, normalized size = 3.05 \begin{align*} \frac{2 \, b^{3} x^{9} + 4 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.95331, size = 63, normalized size = 1.03 \begin{align*} - \frac{a \log{\left (a + b x^{3} \right )}}{b^{4}} - \frac{5 a^{3} + 6 a^{2} b x^{3}}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} + \frac{x^{3}}{3 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09897, size = 84, normalized size = 1.38 \begin{align*} \frac{x^{3}}{3 \, b^{3}} - \frac{a \log \left ({\left | b x^{3} + a \right |}\right )}{b^{4}} + \frac{9 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 4 \, a^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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