Optimal. Leaf size=24 \[ a^2 \log (x)-\frac{a b}{x^2}-\frac{b^2}{4 x^4} \]
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Rubi [A] time = 0.0146094, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ a^2 \log (x)-\frac{a b}{x^2}-\frac{b^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^2}{x} \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{x^3}+\frac{2 a b}{x^2}+\frac{a^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{4 x^4}-\frac{a b}{x^2}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0007909, size = 24, normalized size = 1. \[ a^2 \log (x)-\frac{a b}{x^2}-\frac{b^2}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 23, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{4\,{x}^{4}}}-{\frac{ab}{{x}^{2}}}+{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964707, size = 35, normalized size = 1.46 \begin{align*} \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + b^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47242, size = 62, normalized size = 2.58 \begin{align*} \frac{4 \, a^{2} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - b^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.311263, size = 22, normalized size = 0.92 \begin{align*} a^{2} \log{\left (x \right )} - \frac{4 a b x^{2} + b^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20237, size = 46, normalized size = 1.92 \begin{align*} \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) - \frac{3 \, a^{2} x^{4} + 4 \, a b x^{2} + b^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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