Optimal. Leaf size=77 \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]
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Rubi [A] time = 0.0511226, antiderivative size = 77, 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} \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]
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 )^3}{x^{11}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{11}} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^6} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^{12}+\frac{a^6 b^6}{x^6}+\frac{6 a^5 b^7}{x^5}+\frac{15 a^4 b^8}{x^4}+\frac{20 a^3 b^9}{x^3}+\frac{15 a^2 b^{10}}{x^2}+\frac{6 a b^{11}}{x}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{10 x^{10}}-\frac{3 a^5 b}{4 x^8}-\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}+\frac{b^6 x^2}{2}+6 a b^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0050888, size = 77, normalized size = 1. \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{10\,{x}^{10}}}-{\frac{3\,{a}^{5}b}{4\,{x}^{8}}}-{\frac{5\,{a}^{4}{b}^{2}}{2\,{x}^{6}}}-5\,{\frac{{a}^{3}{b}^{3}}{{x}^{4}}}-{\frac{15\,{a}^{2}{b}^{4}}{2\,{x}^{2}}}+{\frac{{b}^{6}{x}^{2}}{2}}+6\,a{b}^{5}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00177, size = 97, normalized size = 1.26 \begin{align*} \frac{1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac{150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63156, size = 169, normalized size = 2.19 \begin{align*} \frac{10 \, b^{6} x^{12} + 120 \, a b^{5} x^{10} \log \left (x\right ) - 150 \, a^{2} b^{4} x^{8} - 100 \, a^{3} b^{3} x^{6} - 50 \, a^{4} b^{2} x^{4} - 15 \, a^{5} b x^{2} - 2 \, a^{6}}{20 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.621393, size = 73, normalized size = 0.95 \begin{align*} 6 a b^{5} \log{\left (x \right )} + \frac{b^{6} x^{2}}{2} - \frac{2 a^{6} + 15 a^{5} b x^{2} + 50 a^{4} b^{2} x^{4} + 100 a^{3} b^{3} x^{6} + 150 a^{2} b^{4} x^{8}}{20 x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11706, size = 109, normalized size = 1.42 \begin{align*} \frac{1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac{137 \, a b^{5} x^{10} + 150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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