Optimal. Leaf size=34 \[ -\frac{x^{13} f^a \text{Gamma}\left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
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Rubi [A] time = 0.0225019, antiderivative size = 34, normalized size of antiderivative = 1., 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} \[ -\frac{x^{13} f^a \text{Gamma}\left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^2} x^{12} \, dx &=-\frac{f^a x^{13} \Gamma \left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.0056599, size = 34, normalized size = 1. \[ -\frac{x^{13} f^a \text{Gamma}\left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 164, normalized size = 4.8 \begin{align*}{\frac{{f}^{a}{f}^{b{x}^{2}}{x}^{11}}{2\,b\ln \left ( f \right ) }}-{\frac{11\,{f}^{a}{x}^{9}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{99\,{f}^{a}{x}^{7}{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{693\,{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{16\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}}+{\frac{3465\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{32\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}}-{\frac{10395\,{f}^{a}x{f}^{b{x}^{2}}}{64\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}}}+{\frac{10395\,{f}^{a}\sqrt{\pi }}{128\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04814, size = 171, normalized size = 5.03 \begin{align*} \frac{{\left (32 \, b^{5} f^{a} x^{11} \log \left (f\right )^{5} - 176 \, b^{4} f^{a} x^{9} \log \left (f\right )^{4} + 792 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} - 2772 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} + 6930 \, b f^{a} x^{3} \log \left (f\right ) - 10395 \, f^{a} x\right )} f^{b x^{2}}}{64 \, b^{6} \log \left (f\right )^{6}} + \frac{10395 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{128 \, \sqrt{-b \log \left (f\right )} 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 = 1.80379, size = 320, normalized size = 9.41 \begin{align*} -\frac{10395 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) - 2 \,{\left (32 \, b^{6} x^{11} \log \left (f\right )^{6} - 176 \, b^{5} x^{9} \log \left (f\right )^{5} + 792 \, b^{4} x^{7} \log \left (f\right )^{4} - 2772 \, b^{3} x^{5} \log \left (f\right )^{3} + 6930 \, b^{2} x^{3} \log \left (f\right )^{2} - 10395 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{128 \, b^{7} \log \left (f\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20004, size = 157, normalized size = 4.62 \begin{align*} -\frac{10395 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (f\right )} x\right )}{128 \, \sqrt{-b \log \left (f\right )} b^{6} \log \left (f\right )^{6}} + \frac{{\left (32 \, b^{5} x^{11} \log \left (f\right )^{5} - 176 \, b^{4} x^{9} \log \left (f\right )^{4} + 792 \, b^{3} x^{7} \log \left (f\right )^{3} - 2772 \, b^{2} x^{5} \log \left (f\right )^{2} + 6930 \, b x^{3} \log \left (f\right ) - 10395 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{64 \, b^{6} \log \left (f\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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