Optimal. Leaf size=26 \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]
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Rubi [A] time = 0.0122927, antiderivative size = 26, 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} \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^2}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,x^4\right )\\ &=\frac{1}{2} a b x^4+\frac{b^2 x^8}{8}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.000667, size = 26, normalized size = 1. \[ a^2 \log (x)+\frac{1}{2} a b x^4+\frac{b^2 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 0.9 \begin{align*}{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{8}}{8}}+{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 = 0.95713, size = 34, normalized size = 1.31 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + \frac{1}{4} \, a^{2} \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44869, size = 54, normalized size = 2.08 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + 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 = 0.31196, size = 22, normalized size = 0.85 \begin{align*} a^{2} \log{\left (x \right )} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10772, size = 34, normalized size = 1.31 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{2} \, a b x^{4} + \frac{1}{4} \, a^{2} \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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