Optimal. Leaf size=49 \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]
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Rubi [A] time = 0.0356296, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^5} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^5} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^3} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a b^7+\frac{a^4 b^4}{x^3}+\frac{4 a^3 b^5}{x^2}+\frac{6 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{4 x^4}-\frac{2 a^3 b}{x^2}+2 a b^3 x^2+\frac{b^4 x^4}{4}+6 a^2 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0045238, size = 49, normalized size = 1. \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{4\,{x}^{4}}}-2\,{\frac{{a}^{3}b}{{x}^{2}}}+2\,a{b}^{3}{x}^{2}+{\frac{{b}^{4}{x}^{4}}{4}}+6\,{a}^{2}{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 = 0.996828, size = 65, normalized size = 1.33 \begin{align*} \frac{1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac{8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70179, size = 104, normalized size = 2.12 \begin{align*} \frac{b^{4} x^{8} + 8 \, a b^{3} x^{6} + 24 \, a^{2} b^{2} x^{4} \log \left (x\right ) - 8 \, a^{3} b x^{2} - a^{4}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.360037, size = 48, normalized size = 0.98 \begin{align*} 6 a^{2} b^{2} \log{\left (x \right )} + 2 a b^{3} x^{2} + \frac{b^{4} x^{4}}{4} - \frac{a^{4} + 8 a^{3} b x^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18422, size = 80, normalized size = 1.63 \begin{align*} \frac{1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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