Optimal. Leaf size=23 \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]
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Rubi [A] time = 0.0127247, antiderivative size = 23, 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} \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]
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} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,x^2\right )\\ &=a b x^2+\frac{b^2 x^4}{4}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0007734, size = 23, normalized size = 1. \[ a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1. \begin{align*} ab{x}^{2}+{\frac{{b}^{2}{x}^{4}}{4}}+{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 = 1.90964, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{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 = 1.48527, size = 49, normalized size = 2.13 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.244421, size = 20, normalized size = 0.87 \begin{align*} a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50345, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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