Optimal. Leaf size=79 \[ \frac{a^2 x^6}{6 b^3}-\frac{a^3 x^4}{4 b^4}+\frac{a^4 x^2}{2 b^5}-\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]
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Rubi [A] time = 0.0558447, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^6}{6 b^3}-\frac{a^3 x^4}{4 b^4}+\frac{a^4 x^2}{2 b^5}-\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{b^3}-\frac{a x^3}{b^2}+\frac{x^4}{b}-\frac{a^5}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b}-\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}\\ \end{align*}
Mathematica [A] time = 0.005609, size = 79, normalized size = 1. \[ \frac{a^2 x^6}{6 b^3}-\frac{a^3 x^4}{4 b^4}+\frac{a^4 x^2}{2 b^5}-\frac{a^5 \log \left (a+b x^2\right )}{2 b^6}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 68, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{x}^{2}}{2\,{b}^{5}}}-{\frac{{a}^{3}{x}^{4}}{4\,{b}^{4}}}+{\frac{{a}^{2}{x}^{6}}{6\,{b}^{3}}}-{\frac{a{x}^{8}}{8\,{b}^{2}}}+{\frac{{x}^{10}}{10\,b}}-{\frac{{a}^{5}\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 = 1.34039, size = 92, normalized size = 1.16 \begin{align*} -\frac{a^{5} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac{12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24124, size = 153, normalized size = 1.94 \begin{align*} \frac{12 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} + 60 \, a^{4} b x^{2} - 60 \, a^{5} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.319619, size = 68, normalized size = 0.86 \begin{align*} - \frac{a^{5} \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{a^{4} x^{2}}{2 b^{5}} - \frac{a^{3} x^{4}}{4 b^{4}} + \frac{a^{2} x^{6}}{6 b^{3}} - \frac{a x^{8}}{8 b^{2}} + \frac{x^{10}}{10 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.78218, size = 93, normalized size = 1.18 \begin{align*} -\frac{a^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} + \frac{12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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