Optimal. Leaf size=70 \[ \frac{3 a^2 x^2}{2 b^4}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2} \]
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Rubi [A] time = 0.0544227, antiderivative size = 70, 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} \[ \frac{3 a^2 x^2}{2 b^4}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 x^2}{2 b^4}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0231995, size = 60, normalized size = 0.86 \[ \frac{9 a^2 b x^2-\frac{3 a^4}{a+b x^2}-12 a^3 \log \left (a+b x^2\right )-3 a b^2 x^4+b^3 x^6}{6 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 63, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{a{x}^{4}}{2\,{b}^{3}}}+{\frac{{x}^{6}}{6\,{b}^{2}}}-{\frac{{a}^{4}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.62999, size = 88, normalized size = 1.26 \begin{align*} -\frac{a^{4}}{2 \,{\left (b^{6} x^{2} + a b^{5}\right )}} - \frac{2 \, a^{3} \log \left (b x^{2} + a\right )}{b^{5}} + \frac{b^{2} x^{6} - 3 \, a b x^{4} + 9 \, a^{2} x^{2}}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18547, size = 166, normalized size = 2.37 \begin{align*} \frac{b^{4} x^{8} - 2 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} - 3 \, a^{4} - 12 \,{\left (a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{6 \,{\left (b^{6} x^{2} + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.403645, size = 66, normalized size = 0.94 \begin{align*} - \frac{a^{4}}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{2 a^{3} \log{\left (a + b x^{2} \right )}}{b^{5}} + \frac{3 a^{2} x^{2}}{2 b^{4}} - \frac{a x^{4}}{2 b^{3}} + \frac{x^{6}}{6 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.48879, size = 108, normalized size = 1.54 \begin{align*} -\frac{2 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac{b^{4} x^{6} - 3 \, a b^{3} x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, b^{6}} + \frac{4 \, a^{3} b x^{2} + 3 \, a^{4}}{2 \,{\left (b x^{2} + a\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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