Optimal. Leaf size=86 \[ \frac{3 a^2}{2 b^4 \left (a x^2+b\right )}+\frac{a^2}{4 b^3 \left (a x^2+b\right )^2}-\frac{3 a^2 \log \left (a x^2+b\right )}{b^5}+\frac{6 a^2 \log (x)}{b^5}+\frac{3 a}{2 b^4 x^2}-\frac{1}{4 b^3 x^4} \]
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Rubi [A] time = 0.0543483, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ \frac{3 a^2}{2 b^4 \left (a x^2+b\right )}+\frac{a^2}{4 b^3 \left (a x^2+b\right )^2}-\frac{3 a^2 \log \left (a x^2+b\right )}{b^5}+\frac{6 a^2 \log (x)}{b^5}+\frac{3 a}{2 b^4 x^2}-\frac{1}{4 b^3 x^4} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^{11}} \, dx &=\int \frac{1}{x^5 \left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^3}-\frac{3 a}{b^4 x^2}+\frac{6 a^2}{b^5 x}-\frac{a^3}{b^3 (b+a x)^3}-\frac{3 a^3}{b^4 (b+a x)^2}-\frac{6 a^3}{b^5 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 b^3 x^4}+\frac{3 a}{2 b^4 x^2}+\frac{a^2}{4 b^3 \left (b+a x^2\right )^2}+\frac{3 a^2}{2 b^4 \left (b+a x^2\right )}+\frac{6 a^2 \log (x)}{b^5}-\frac{3 a^2 \log \left (b+a x^2\right )}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0441437, size = 74, normalized size = 0.86 \[ \frac{\frac{b \left (18 a^2 b x^4+12 a^3 x^6+4 a b^2 x^2-b^3\right )}{x^4 \left (a x^2+b\right )^2}-12 a^2 \log \left (a x^2+b\right )+24 a^2 \log (x)}{4 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{b}^{3}{x}^{4}}}+{\frac{3\,a}{2\,{b}^{4}{x}^{2}}}+{\frac{{a}^{2}}{4\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( a{x}^{2}+b \right ) }}+6\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{5}}}-3\,{\frac{{a}^{2}\ln \left ( a{x}^{2}+b \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00575, size = 124, normalized size = 1.44 \begin{align*} \frac{12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \,{\left (a^{2} b^{4} x^{8} + 2 \, a b^{5} x^{6} + b^{6} x^{4}\right )}} - \frac{3 \, a^{2} \log \left (a x^{2} + b\right )}{b^{5}} + \frac{3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42516, size = 274, normalized size = 3.19 \begin{align*} \frac{12 \, a^{3} b x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a b^{3} x^{2} - b^{4} - 12 \,{\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (a x^{2} + b\right ) + 24 \,{\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{5} x^{8} + 2 \, a b^{6} x^{6} + b^{7} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35903, size = 90, normalized size = 1.05 \begin{align*} \frac{6 a^{2} \log{\left (x \right )}}{b^{5}} - \frac{3 a^{2} \log{\left (x^{2} + \frac{b}{a} \right )}}{b^{5}} + \frac{12 a^{3} x^{6} + 18 a^{2} b x^{4} + 4 a b^{2} x^{2} - b^{3}}{4 a^{2} b^{4} x^{8} + 8 a b^{5} x^{6} + 4 b^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19774, size = 108, normalized size = 1.26 \begin{align*} \frac{3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} - \frac{3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{b^{5}} + \frac{12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \,{\left (a x^{4} + b x^{2}\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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