Optimal. Leaf size=87 \[ \frac{3 b^2 x^2}{a^5}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}+\frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]
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Rubi [A] time = 0.0639311, antiderivative size = 87, 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} \[ \frac{3 b^2 x^2}{a^5}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}+\frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]
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^2}\right )^3} \, dx &=\int \frac{x^{11}}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{6 b^2}{a^5}-\frac{3 b x}{a^4}+\frac{x^2}{a^3}-\frac{b^5}{a^5 (b+a x)^3}+\frac{5 b^4}{a^5 (b+a x)^2}-\frac{10 b^3}{a^5 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 b^2 x^2}{a^5}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3}+\frac{b^5}{4 a^6 \left (b+a x^2\right )^2}-\frac{5 b^4}{2 a^6 \left (b+a x^2\right )}-\frac{5 b^3 \log \left (b+a x^2\right )}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0466435, size = 71, normalized size = 0.82 \[ \frac{-9 a^2 b x^4+2 a^3 x^6+36 a b^2 x^2-\frac{3 b^4 \left (10 a x^2+9 b\right )}{\left (a x^2+b\right )^2}-60 b^3 \log \left (a x^2+b\right )}{12 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 80, normalized size = 0.9 \begin{align*} 3\,{\frac{{b}^{2}{x}^{2}}{{a}^{5}}}-{\frac{3\,b{x}^{4}}{4\,{a}^{4}}}+{\frac{{x}^{6}}{6\,{a}^{3}}}+{\frac{{b}^{5}}{4\,{a}^{6} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{5\,{b}^{4}}{2\,{a}^{6} \left ( a{x}^{2}+b \right ) }}-5\,{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01067, size = 120, normalized size = 1.38 \begin{align*} -\frac{10 \, a b^{4} x^{2} + 9 \, b^{5}}{4 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} - \frac{5 \, b^{3} \log \left (a x^{2} + b\right )}{a^{6}} + \frac{2 \, a^{2} x^{6} - 9 \, a b x^{4} + 36 \, b^{2} x^{2}}{12 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42239, size = 240, normalized size = 2.76 \begin{align*} \frac{2 \, a^{5} x^{10} - 5 \, a^{4} b x^{8} + 20 \, a^{3} b^{2} x^{6} + 63 \, a^{2} b^{3} x^{4} + 6 \, a b^{4} x^{2} - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )} \log \left (a x^{2} + b\right )}{12 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.796733, size = 90, normalized size = 1.03 \begin{align*} - \frac{10 a b^{4} x^{2} + 9 b^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{2} + 4 a^{6} b^{2}} + \frac{x^{6}}{6 a^{3}} - \frac{3 b x^{4}}{4 a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} - \frac{5 b^{3} \log{\left (a x^{2} + b \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18754, size = 124, normalized size = 1.43 \begin{align*} -\frac{5 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{6}} + \frac{30 \, a^{2} b^{3} x^{4} + 50 \, a b^{4} x^{2} + 21 \, b^{5}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x^{6} - 9 \, a^{5} b x^{4} + 36 \, a^{4} b^{2} x^{2}}{12 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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