Optimal. Leaf size=91 \[ \frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4} \]
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Rubi [A] time = 0.0926364, antiderivative size = 91, 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} \[ \frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^{11}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \frac{x^5}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \left (-\frac{4 a}{b^9}+\frac{x}{b^8}-\frac{a^5}{b^9 (a+b x)^4}+\frac{5 a^4}{b^9 (a+b x)^3}-\frac{10 a^3}{b^9 (a+b x)^2}+\frac{10 a^2}{b^9 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4}+\frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0305118, size = 78, normalized size = 0.86 \[ \frac{\frac{2 a^5}{\left (a+b x^2\right )^3}-\frac{15 a^4}{\left (a+b x^2\right )^2}+\frac{60 a^3}{a+b x^2}+60 a^2 \log \left (a+b x^2\right )-24 a b x^2+3 b^2 x^4}{12 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 86, normalized size = 1. \begin{align*} -2\,{\frac{a{x}^{2}}{{b}^{5}}}+{\frac{{x}^{4}}{4\,{b}^{4}}}+{\frac{{a}^{5}}{6\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{5\,{a}^{4}}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+5\,{\frac{{a}^{3}}{{b}^{6} \left ( b{x}^{2}+a \right ) }}+5\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00571, size = 134, normalized size = 1.47 \begin{align*} \frac{60 \, a^{3} b^{2} x^{4} + 105 \, a^{4} b x^{2} + 47 \, a^{5}}{12 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} + \frac{5 \, a^{2} \log \left (b x^{2} + a\right )}{b^{6}} + \frac{b x^{4} - 8 \, a x^{2}}{4 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63722, size = 285, normalized size = 3.13 \begin{align*} \frac{3 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} - 63 \, a^{2} b^{3} x^{6} - 9 \, a^{3} b^{2} x^{4} + 81 \, a^{4} b x^{2} + 47 \, a^{5} + 60 \,{\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.784291, size = 100, normalized size = 1.1 \begin{align*} \frac{5 a^{2} \log{\left (a + b x^{2} \right )}}{b^{6}} - \frac{2 a x^{2}}{b^{5}} + \frac{47 a^{5} + 105 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4}}{12 a^{3} b^{6} + 36 a^{2} b^{7} x^{2} + 36 a b^{8} x^{4} + 12 b^{9} x^{6}} + \frac{x^{4}}{4 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13509, size = 123, normalized size = 1.35 \begin{align*} \frac{5 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{b^{4} x^{4} - 8 \, a b^{3} x^{2}}{4 \, b^{8}} - \frac{110 \, a^{2} b^{3} x^{6} + 270 \, a^{3} b^{2} x^{4} + 225 \, a^{4} b x^{2} + 63 \, a^{5}}{12 \,{\left (b x^{2} + a\right )}^{3} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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