Optimal. Leaf size=56 \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]
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Rubi [A] time = 0.0401671, antiderivative size = 56, 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} \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+\frac{b}{x^3}\right )^2} \, dx &=\int \frac{x^{11}}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(b+a x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{2 b}{a^3}+\frac{x}{a^2}-\frac{b^3}{a^3 (b+a x)^2}+\frac{3 b^2}{a^3 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2}+\frac{b^3}{3 a^4 \left (b+a x^3\right )}+\frac{b^2 \log \left (b+a x^3\right )}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0178297, size = 49, normalized size = 0.88 \[ \frac{a^2 x^6+\frac{2 b^3}{a x^3+b}+6 b^2 \log \left (a x^3+b\right )-4 a b x^3}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 51, normalized size = 0.9 \begin{align*} -{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{3}}{3\,{a}^{4} \left ( a{x}^{3}+b \right ) }}+{\frac{{b}^{2}\ln \left ( a{x}^{3}+b \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00343, size = 72, normalized size = 1.29 \begin{align*} \frac{b^{3}}{3 \,{\left (a^{5} x^{3} + a^{4} b\right )}} + \frac{b^{2} \log \left (a x^{3} + b\right )}{a^{4}} + \frac{a x^{6} - 4 \, b x^{3}}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42172, size = 143, normalized size = 2.55 \begin{align*} \frac{a^{3} x^{9} - 3 \, a^{2} b x^{6} - 4 \, a b^{2} x^{3} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{3} + b^{3}\right )} \log \left (a x^{3} + b\right )}{6 \,{\left (a^{5} x^{3} + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.604334, size = 53, normalized size = 0.95 \begin{align*} \frac{b^{3}}{3 a^{5} x^{3} + 3 a^{4} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{b^{2} \log{\left (a x^{3} + b \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20849, size = 73, normalized size = 1.3 \begin{align*} \frac{b^{2} \log \left ({\left | a x^{3} + b \right |}\right )}{a^{4}} + \frac{b^{3}}{3 \,{\left (a x^{3} + b\right )} a^{4}} + \frac{a^{2} x^{6} - 4 \, a b x^{3}}{6 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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