Optimal. Leaf size=83 \[ \frac{3 a^2 x^4}{4 b^4}-\frac{2 a^3 x^2}{b^5}+\frac{a^5}{2 b^6 \left (a+b x^2\right )}+\frac{5 a^4 \log \left (a+b x^2\right )}{2 b^6}-\frac{a x^6}{3 b^3}+\frac{x^8}{8 b^2} \]
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Rubi [A] time = 0.0821507, antiderivative size = 83, 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} \[ \frac{3 a^2 x^4}{4 b^4}-\frac{2 a^3 x^2}{b^5}+\frac{a^5}{2 b^6 \left (a+b x^2\right )}+\frac{5 a^4 \log \left (a+b x^2\right )}{2 b^6}-\frac{a x^6}{3 b^3}+\frac{x^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{x^{11}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{x^5}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{b^7}+\frac{3 a^2 x}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^3}{b^4}-\frac{a^5}{b^7 (a+b x)^2}+\frac{5 a^4}{b^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a^3 x^2}{b^5}+\frac{3 a^2 x^4}{4 b^4}-\frac{a x^6}{3 b^3}+\frac{x^8}{8 b^2}+\frac{a^5}{2 b^6 \left (a+b x^2\right )}+\frac{5 a^4 \log \left (a+b x^2\right )}{2 b^6}\\ \end{align*}
Mathematica [A] time = 0.0203641, size = 72, normalized size = 0.87 \[ \frac{18 a^2 b^2 x^4-48 a^3 b x^2+\frac{12 a^5}{a+b x^2}+60 a^4 \log \left (a+b x^2\right )-8 a b^3 x^6+3 b^4 x^8}{24 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 74, normalized size = 0.9 \begin{align*} -2\,{\frac{{x}^{2}{a}^{3}}{{b}^{5}}}+{\frac{3\,{a}^{2}{x}^{4}}{4\,{b}^{4}}}-{\frac{a{x}^{6}}{3\,{b}^{3}}}+{\frac{{x}^{8}}{8\,{b}^{2}}}+{\frac{{a}^{5}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{4}\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 = 1.18691, size = 104, normalized size = 1.25 \begin{align*} \frac{a^{5}}{2 \,{\left (b^{7} x^{2} + a b^{6}\right )}} + \frac{5 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac{3 \, b^{3} x^{8} - 8 \, a b^{2} x^{6} + 18 \, a^{2} b x^{4} - 48 \, a^{3} x^{2}}{24 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65008, size = 198, normalized size = 2.39 \begin{align*} \frac{3 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} - 48 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \,{\left (a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{7} x^{2} + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.440946, size = 80, normalized size = 0.96 \begin{align*} \frac{a^{5}}{2 a b^{6} + 2 b^{7} x^{2}} + \frac{5 a^{4} \log{\left (a + b x^{2} \right )}}{2 b^{6}} - \frac{2 a^{3} x^{2}}{b^{5}} + \frac{3 a^{2} x^{4}}{4 b^{4}} - \frac{a x^{6}}{3 b^{3}} + \frac{x^{8}}{8 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20629, size = 124, normalized size = 1.49 \begin{align*} \frac{5 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{5 \, a^{4} b x^{2} + 4 \, a^{5}}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{8} - 8 \, a b^{5} x^{6} + 18 \, a^{2} b^{4} x^{4} - 48 \, a^{3} b^{3} x^{2}}{24 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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