Optimal. Leaf size=49 \[ -\frac{3 a^2 b^2}{x^2}-\frac{a^3 b}{x^4}-\frac{a^4}{6 x^6}+4 a b^3 \log (x)+\frac{b^4 x^2}{2} \]
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Rubi [A] time = 0.0326376, 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} \[ -\frac{3 a^2 b^2}{x^2}-\frac{a^3 b}{x^4}-\frac{a^4}{6 x^6}+4 a b^3 \log (x)+\frac{b^4 x^2}{2} \]
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^7} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^7} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^4} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^8+\frac{a^4 b^4}{x^4}+\frac{4 a^3 b^5}{x^3}+\frac{6 a^2 b^6}{x^2}+\frac{4 a b^7}{x}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{6 x^6}-\frac{a^3 b}{x^4}-\frac{3 a^2 b^2}{x^2}+\frac{b^4 x^2}{2}+4 a b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0045966, size = 49, normalized size = 1. \[ -\frac{3 a^2 b^2}{x^2}-\frac{a^3 b}{x^4}-\frac{a^4}{6 x^6}+4 a b^3 \log (x)+\frac{b^4 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{6\,{x}^{6}}}-{\frac{{a}^{3}b}{{x}^{4}}}-3\,{\frac{{b}^{2}{a}^{2}}{{x}^{2}}}+{\frac{{b}^{4}{x}^{2}}{2}}+4\,a{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9814, size = 65, normalized size = 1.33 \begin{align*} \frac{1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6557, size = 108, normalized size = 2.2 \begin{align*} \frac{3 \, b^{4} x^{8} + 24 \, a b^{3} x^{6} \log \left (x\right ) - 18 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x^{2} - a^{4}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.419184, size = 48, normalized size = 0.98 \begin{align*} 4 a b^{3} \log{\left (x \right )} + \frac{b^{4} x^{2}}{2} - \frac{a^{4} + 6 a^{3} b x^{2} + 18 a^{2} b^{2} x^{4}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10848, size = 77, normalized size = 1.57 \begin{align*} \frac{1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac{22 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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