Optimal. Leaf size=27 \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]
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Rubi [A] time = 0.0135157, antiderivative size = 27, 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} \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{2 x^2}+\frac{b^2 x^2}{2}+2 a b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0007728, size = 27, normalized size = 1. \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}+{\frac{{b}^{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 = 1.33831, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39969, size = 59, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.270194, size = 24, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.55442, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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