Optimal. Leaf size=109 \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]
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Rubi [A] time = 0.102303, antiderivative size = 109, 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} \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]
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 )^3} \, dx &=b^6 \int \frac{x^{11}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{x^5}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \left (-\frac{a^5}{b^{11} (a+b x)^6}+\frac{5 a^4}{b^{11} (a+b x)^5}-\frac{10 a^3}{b^{11} (a+b x)^4}+\frac{10 a^2}{b^{11} (a+b x)^3}-\frac{5 a}{b^{11} (a+b x)^2}+\frac{1}{b^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6}\\ \end{align*}
Mathematica [A] time = 0.0232354, size = 72, normalized size = 0.66 \[ \frac{\frac{a \left (1100 a^2 b^2 x^4+625 a^3 b x^2+137 a^4+900 a b^3 x^6+300 b^4 x^8\right )}{\left (a+b x^2\right )^5}+60 \log \left (a+b x^2\right )}{120 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 98, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}}{10\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{5\,{a}^{4}}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{5\,{a}^{3}}{3\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{5\,{a}^{2}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,a}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20553, size = 163, normalized size = 1.5 \begin{align*} \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5}}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69063, size = 367, normalized size = 3.37 \begin{align*} \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5} + 60 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.26721, size = 124, normalized size = 1.14 \begin{align*} \frac{137 a^{5} + 625 a^{4} b x^{2} + 1100 a^{3} b^{2} x^{4} + 900 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 a^{5} b^{6} + 600 a^{4} b^{7} x^{2} + 1200 a^{3} b^{8} x^{4} + 1200 a^{2} b^{9} x^{6} + 600 a b^{10} x^{8} + 120 b^{11} x^{10}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13556, size = 101, normalized size = 0.93 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{137 \, b^{4} x^{10} + 385 \, a b^{3} x^{8} + 470 \, a^{2} b^{2} x^{6} + 270 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \,{\left (b x^{2} + a\right )}^{5} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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