Optimal. Leaf size=34 \[ \frac{1}{2} x^9 f^a \left (-\frac{b \log (f)}{x^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \]
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Rubi [A] time = 0.0236792, 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} \[ \frac{1}{2} x^9 f^a \left (-\frac{b \log (f)}{x^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+\frac{b}{x^2}} x^8 \, dx &=\frac{1}{2} f^a x^9 \Gamma \left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \left (-\frac{b \log (f)}{x^2}\right )^{9/2}\\ \end{align*}
Mathematica [A] time = 0.004138, size = 34, normalized size = 1. \[ \frac{1}{2} x^9 f^a \left (-\frac{b \log (f)}{x^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 133, normalized size = 3.9 \begin{align*}{\frac{{f}^{a}{x}^{9}}{9}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{7}}{63}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{5}}{315}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{3}}{945}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}x}{945}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}\sqrt{\pi }}{945}{\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 = 1.20365, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{2} \, f^{a} x^{9} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{9}{2}} \Gamma \left (-\frac{9}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75018, size = 259, normalized size = 7.62 \begin{align*} \frac{16}{945} \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b^{4} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{4} + \frac{1}{945} \,{\left (105 \, x^{9} + 30 \, b x^{7} \log \left (f\right ) + 12 \, b^{2} x^{5} \log \left (f\right )^{2} + 8 \, b^{3} x^{3} \log \left (f\right )^{3} + 16 \, b^{4} x \log \left (f\right )^{4}\right )} f^{\frac{a x^{2} + b}{x^{2}}} \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 f^{a + \frac{b}{x^{2}}} x^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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