Optimal. Leaf size=70 \[ \frac{3 b^2 x^2}{2 a^4}-\frac{b^4}{2 a^5 \left (a x^2+b\right )}-\frac{2 b^3 \log \left (a x^2+b\right )}{a^5}-\frac{b x^4}{2 a^3}+\frac{x^6}{6 a^2} \]
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Rubi [A] time = 0.0521735, antiderivative size = 70, 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} \[ \frac{3 b^2 x^2}{2 a^4}-\frac{b^4}{2 a^5 \left (a x^2+b\right )}-\frac{2 b^3 \log \left (a x^2+b\right )}{a^5}-\frac{b x^4}{2 a^3}+\frac{x^6}{6 a^2} \]
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^2}\right )^2} \, dx &=\int \frac{x^9}{\left (b+a x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{3 b^2}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{a^2}+\frac{b^4}{a^4 (b+a x)^2}-\frac{4 b^3}{a^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 b^2 x^2}{2 a^4}-\frac{b x^4}{2 a^3}+\frac{x^6}{6 a^2}-\frac{b^4}{2 a^5 \left (b+a x^2\right )}-\frac{2 b^3 \log \left (b+a x^2\right )}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0206056, size = 60, normalized size = 0.86 \[ \frac{-3 a^2 b x^4+a^3 x^6+9 a b^2 x^2-\frac{3 b^4}{a x^2+b}-12 b^3 \log \left (a x^2+b\right )}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 63, normalized size = 0.9 \begin{align*}{\frac{3\,{b}^{2}{x}^{2}}{2\,{a}^{4}}}-{\frac{b{x}^{4}}{2\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}-{\frac{{b}^{4}}{2\,{a}^{5} \left ( a{x}^{2}+b \right ) }}-2\,{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986552, size = 88, normalized size = 1.26 \begin{align*} -\frac{b^{4}}{2 \,{\left (a^{6} x^{2} + a^{5} b\right )}} - \frac{2 \, b^{3} \log \left (a x^{2} + b\right )}{a^{5}} + \frac{a^{2} x^{6} - 3 \, a b x^{4} + 9 \, b^{2} x^{2}}{6 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47581, size = 166, normalized size = 2.37 \begin{align*} \frac{a^{4} x^{8} - 2 \, a^{3} b x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a b^{3} x^{2} - 3 \, b^{4} - 12 \,{\left (a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{6 \,{\left (a^{6} x^{2} + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.552065, size = 66, normalized size = 0.94 \begin{align*} - \frac{b^{4}}{2 a^{6} x^{2} + 2 a^{5} b} + \frac{x^{6}}{6 a^{2}} - \frac{b x^{4}}{2 a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{2 b^{3} \log{\left (a x^{2} + b \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15216, size = 108, normalized size = 1.54 \begin{align*} -\frac{2 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac{a^{4} x^{6} - 3 \, a^{3} b x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, a^{6}} + \frac{4 \, a b^{3} x^{2} + 3 \, b^{4}}{2 \,{\left (a x^{2} + b\right )} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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