Optimal. Leaf size=65 \[ \frac{b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac{3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac{3 b \log \left (a x^2+b\right )}{2 a^4}+\frac{x^2}{2 a^3} \]
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Rubi [A] time = 0.0456249, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ \frac{b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac{3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac{3 b \log \left (a x^2+b\right )}{2 a^4}+\frac{x^2}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^7}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{b^3}{a^3 (b+a x)^3}+\frac{3 b^2}{a^3 (b+a x)^2}-\frac{3 b}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 a^3}+\frac{b^3}{4 a^4 \left (b+a x^2\right )^2}-\frac{3 b^2}{2 a^4 \left (b+a x^2\right )}-\frac{3 b \log \left (b+a x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0537783, size = 48, normalized size = 0.74 \[ -\frac{\frac{b^2 \left (6 a x^2+5 b\right )}{\left (a x^2+b\right )^2}+6 b \log \left (a x^2+b\right )-2 a x^2}{4 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 58, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{3}}}+{\frac{{b}^{3}}{4\,{a}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,b\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988962, size = 89, normalized size = 1.37 \begin{align*} -\frac{6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} + \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, b \log \left (a x^{2} + b\right )}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44664, size = 186, normalized size = 2.86 \begin{align*} \frac{2 \, a^{3} x^{6} + 4 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} - 5 \, b^{3} - 6 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.714798, size = 66, normalized size = 1.02 \begin{align*} - \frac{6 a b^{2} x^{2} + 5 b^{3}}{4 a^{6} x^{4} + 8 a^{5} b x^{2} + 4 a^{4} b^{2}} + \frac{x^{2}}{2 a^{3}} - \frac{3 b \log{\left (a x^{2} + b \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27198, size = 72, normalized size = 1.11 \begin{align*} \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} - \frac{6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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