Optimal. Leaf size=86 \[ -\frac {3 a^2 \log \left (a x^2+b\right )}{b^5}+\frac {6 a^2 \log (x)}{b^5}+\frac {3 a^2}{2 b^4 \left (a x^2+b\right )}+\frac {a^2}{4 b^3 \left (a x^2+b\right )^2}+\frac {3 a}{2 b^4 x^2}-\frac {1}{4 b^3 x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 44
Rule 263
Rule 266
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*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.86 \[ \frac {-12 a^2 \log \left (a x^2+b\right )+24 a^2 \log (x)+\frac {b \left (12 a^3 x^6+18 a^2 b x^4+4 a b^2 x^2-b^3\right )}{x^4 \left (a x^2+b\right )^2}}{4 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 134, normalized size = 1.56 \[ \frac {12 \, a^{3} b x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a b^{3} x^{2} - b^{4} - 12 \, {\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (a x^{2} + b\right ) + 24 \, {\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{2} b^{5} x^{8} + 2 \, a b^{6} x^{6} + b^{7} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 80, normalized size = 0.93 \[ \frac {3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} - \frac {3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{b^{5}} + \frac {12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \, {\left (a x^{4} + b x^{2}\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 0.92 \[ \frac {a^{2}}{4 \left (a \,x^{2}+b \right )^{2} b^{3}}+\frac {3 a^{2}}{2 \left (a \,x^{2}+b \right ) b^{4}}+\frac {6 a^{2} \ln \relax (x )}{b^{5}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{b^{5}}+\frac {3 a}{2 b^{4} x^{2}}-\frac {1}{4 b^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 92, normalized size = 1.07 \[ \frac {12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \, {\left (a^{2} b^{4} x^{8} + 2 \, a b^{5} x^{6} + b^{6} x^{4}\right )}} - \frac {3 \, a^{2} \log \left (a x^{2} + b\right )}{b^{5}} + \frac {3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 88, normalized size = 1.02 \[ \frac {\frac {a\,x^2}{b^2}-\frac {1}{4\,b}+\frac {9\,a^2\,x^4}{2\,b^3}+\frac {3\,a^3\,x^6}{b^4}}{a^2\,x^8+2\,a\,b\,x^6+b^2\,x^4}-\frac {3\,a^2\,\ln \left (a\,x^2+b\right )}{b^5}+\frac {6\,a^2\,\ln \relax (x)}{b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 90, normalized size = 1.05 \[ \frac {6 a^{2} \log {\relax (x )}}{b^{5}} - \frac {3 a^{2} \log {\left (x^{2} + \frac {b}{a} \right )}}{b^{5}} + \frac {12 a^{3} x^{6} + 18 a^{2} b x^{4} + 4 a b^{2} x^{2} - b^{3}}{4 a^{2} b^{4} x^{8} + 8 a b^{5} x^{6} + 4 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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