Optimal. Leaf size=78 \[ -\frac{f^{a+b x^2} \left (-b^5 x^{10} \log ^5(f)+5 b^4 x^8 \log ^4(f)-20 b^3 x^6 \log ^3(f)+60 b^2 x^4 \log ^2(f)-120 b x^2 \log (f)+120\right )}{2 b^6 \log ^6(f)} \]
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Rubi [C] time = 0.0254629, antiderivative size = 24, normalized size of antiderivative = 0.31, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac{f^a \text{Gamma}\left (6,-b x^2 \log (f)\right )}{2 b^6 \log ^6(f)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^2} x^{11} \, dx &=-\frac{f^a \Gamma \left (6,-b x^2 \log (f)\right )}{2 b^6 \log ^6(f)}\\ \end{align*}
Mathematica [C] time = 0.0030158, size = 24, normalized size = 0.31 \[ -\frac{f^a \text{Gamma}\left (6,-b x^2 \log (f)\right )}{2 b^6 \log ^6(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 76, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{5}{x}^{10} \left ( \ln \left ( f \right ) \right ) ^{5}-5\,{b}^{4}{x}^{8} \left ( \ln \left ( f \right ) \right ) ^{4}+20\,{b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}-60\,{b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}+120\,b{x}^{2}\ln \left ( f \right ) -120 \right ){f}^{b{x}^{2}+a}}{2\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16641, size = 124, normalized size = 1.59 \begin{align*} \frac{{\left (b^{5} f^{a} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} f^{a} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} f^{a} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} + 120 \, b f^{a} x^{2} \log \left (f\right ) - 120 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{6} \log \left (f\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54697, size = 194, normalized size = 2.49 \begin{align*} \frac{{\left (b^{5} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} x^{4} \log \left (f\right )^{2} + 120 \, b x^{2} \log \left (f\right ) - 120\right )} f^{b x^{2} + a}}{2 \, b^{6} \log \left (f\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.16055, size = 95, normalized size = 1.22 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}} \left (b^{5} x^{10} \log{\left (f \right )}^{5} - 5 b^{4} x^{8} \log{\left (f \right )}^{4} + 20 b^{3} x^{6} \log{\left (f \right )}^{3} - 60 b^{2} x^{4} \log{\left (f \right )}^{2} + 120 b x^{2} \log{\left (f \right )} - 120\right )}{2 b^{6} \log{\left (f \right )}^{6}} & \text{for}\: 2 b^{6} \log{\left (f \right )}^{6} \neq 0 \\\frac{x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32157, size = 107, normalized size = 1.37 \begin{align*} \frac{{\left (b^{5} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} x^{4} \log \left (f\right )^{2} + 120 \, b x^{2} \log \left (f\right ) - 120\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{6} \log \left (f\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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