Optimal. Leaf size=23 \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]
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Rubi [A] time = 0.0139885, antiderivative size = 23, 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} \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]
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^3 \, dx &=\int \frac{\left (b+a x^2\right )^2}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (2 a b+\frac{b^2}{x}+a^2 x\right ) \, dx,x,x^2\right )\\ &=a b x^2+\frac{a^2 x^4}{4}+b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0010965, size = 23, normalized size = 1. \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 1. \begin{align*} ab{x}^{2}+{\frac{{a}^{2}{x}^{4}}{4}}+{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969425, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43169, size = 49, normalized size = 2.13 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + b^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.246183, size = 20, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16635, size = 30, normalized size = 1.3 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + b^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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