Optimal. Leaf size=34 \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]
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Rubi [A] time = 0.0256291, 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} \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{f^{a+\frac{b}{x^2}}}{x^{12}} \, dx &=\frac{f^a \Gamma \left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0052913, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 146, normalized size = 4.3 \begin{align*} -{\frac{{f}^{a}}{2\,b{x}^{9}\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{9\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{7}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{63\,{f}^{a}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{5}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{315\,{f}^{a}}{16\,{b}^{4}{x}^{3} \left ( \ln \left ( f \right ) \right ) ^{4}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{945\,{f}^{a}}{32\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}x}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{945\,{f}^{a}\sqrt{\pi }}{64\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \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 = 1.25357, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{11}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{11} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78804, size = 293, normalized size = 8.62 \begin{align*} -\frac{945 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x^{9} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + 2 \,{\left (945 \, b x^{8} \log \left (f\right ) - 630 \, b^{2} x^{6} \log \left (f\right )^{2} + 252 \, b^{3} x^{4} \log \left (f\right )^{3} - 72 \, b^{4} x^{2} \log \left (f\right )^{4} + 16 \, b^{5} \log \left (f\right )^{5}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{64 \, b^{6} x^{9} \log \left (f\right )^{6}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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