Optimal. Leaf size=27 \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]
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Rubi [A] time = 0.0144858, antiderivative size = 27, 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} \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^2 x \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{2 x^2}+\frac{a^2 x^2}{2}+2 a b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0044463, size = 27, normalized size = 1. \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{2\,{x}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998854, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4484, size = 59, normalized size = 2.19 \begin{align*} \frac{a^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.27389, size = 24, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{2}}{2} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12994, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac{b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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