Optimal. Leaf size=76 \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]
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Rubi [A] time = 0.0486256, antiderivative size = 76, 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} \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]
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^{13}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{13}} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^7} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^6 b^6}{x^7}+\frac{6 a^5 b^7}{x^6}+\frac{15 a^4 b^8}{x^5}+\frac{20 a^3 b^9}{x^4}+\frac{15 a^2 b^{10}}{x^3}+\frac{6 a b^{11}}{x^2}+\frac{b^{12}}{x}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.004745, size = 76, normalized size = 1. \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{12\,{x}^{12}}}-{\frac{3\,{a}^{5}b}{5\,{x}^{10}}}-{\frac{15\,{a}^{4}{b}^{2}}{8\,{x}^{8}}}-{\frac{10\,{a}^{3}{b}^{3}}{3\,{x}^{6}}}-{\frac{15\,{a}^{2}{b}^{4}}{4\,{x}^{4}}}-3\,{\frac{a{b}^{5}}{{x}^{2}}}+{b}^{6}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982927, size = 97, normalized size = 1.28 \begin{align*} \frac{1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac{360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59287, size = 174, normalized size = 2.29 \begin{align*} \frac{120 \, b^{6} x^{12} \log \left (x\right ) - 360 \, a b^{5} x^{10} - 450 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 225 \, a^{4} b^{2} x^{4} - 72 \, a^{5} b x^{2} - 10 \, a^{6}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.718432, size = 71, normalized size = 0.93 \begin{align*} b^{6} \log{\left (x \right )} - \frac{10 a^{6} + 72 a^{5} b x^{2} + 225 a^{4} b^{2} x^{4} + 400 a^{3} b^{3} x^{6} + 450 a^{2} b^{4} x^{8} + 360 a b^{5} x^{10}}{120 x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15707, size = 108, normalized size = 1.42 \begin{align*} \frac{1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac{147 \, b^{6} x^{12} + 360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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