Optimal. Leaf size=44 \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]
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Rubi [A] time = 0.0312199, antiderivative size = 44, 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} \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]
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 )^2} \, dx &=\int \frac{x^5}{\left (b+a x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2}+\frac{b^2}{a^2 (b+a x)^2}-\frac{2 b}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 a^2}-\frac{b^2}{2 a^3 \left (b+a x^2\right )}-\frac{b \log \left (b+a x^2\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0167486, size = 38, normalized size = 0.86 \[ \frac{-\frac{b^2}{a x^2+b}-2 b \log \left (a x^2+b\right )+a x^2}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( a{x}^{2}+b \right ) }}-{\frac{b\ln \left ( a{x}^{2}+b \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987699, size = 58, normalized size = 1.32 \begin{align*} -\frac{b^{2}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} + \frac{x^{2}}{2 \, a^{2}} - \frac{b \log \left (a x^{2} + b\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42715, size = 113, normalized size = 2.57 \begin{align*} \frac{a^{2} x^{4} + a b x^{2} - b^{2} - 2 \,{\left (a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.483175, size = 39, normalized size = 0.89 \begin{align*} - \frac{b^{2}}{2 a^{4} x^{2} + 2 a^{3} b} + \frac{x^{2}}{2 a^{2}} - \frac{b \log{\left (a x^{2} + b \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22766, size = 55, normalized size = 1.25 \begin{align*} \frac{x^{2}}{2 \, a^{2}} - \frac{b \log \left ({\left | a x^{2} + b \right |}\right )}{a^{3}} - \frac{b^{2}}{2 \,{\left (a x^{2} + b\right )} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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