Optimal. Leaf size=65 \[ \frac{f^{a+b x^2} \left (b^4 x^8 \log ^4(f)-4 b^3 x^6 \log ^3(f)+12 b^2 x^4 \log ^2(f)-24 b x^2 \log (f)+24\right )}{2 b^5 \log ^5(f)} \]
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Rubi [C] time = 0.0240763, antiderivative size = 24, normalized size of antiderivative = 0.37, 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} \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^2} x^9 \, dx &=\frac{f^a \Gamma \left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)}\\ \end{align*}
Mathematica [C] time = 0.0028788, size = 24, normalized size = 0.37 \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 64, normalized size = 1. \begin{align*}{\frac{{f}^{b{x}^{2}+a} \left ( 24-24\,b{x}^{2}\ln \left ( f \right ) +12\,{b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}-4\,{b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}+{b}^{4}{x}^{8} \left ( \ln \left ( f \right ) \right ) ^{4} \right ) }{2\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12678, size = 104, normalized size = 1.6 \begin{align*} \frac{{\left (b^{4} f^{a} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} f^{a} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} - 24 \, b f^{a} x^{2} \log \left (f\right ) + 24 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.528, size = 161, normalized size = 2.48 \begin{align*} \frac{{\left (b^{4} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} - 24 \, b x^{2} \log \left (f\right ) + 24\right )} f^{b x^{2} + a}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.253196, size = 82, normalized size = 1.26 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}} \left (b^{4} x^{8} \log{\left (f \right )}^{4} - 4 b^{3} x^{6} \log{\left (f \right )}^{3} + 12 b^{2} x^{4} \log{\left (f \right )}^{2} - 24 b x^{2} \log{\left (f \right )} + 24\right )}{2 b^{5} \log{\left (f \right )}^{5}} & \text{for}\: 2 b^{5} \log{\left (f \right )}^{5} \neq 0 \\\frac{x^{10}}{10} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25853, size = 90, normalized size = 1.38 \begin{align*} \frac{{\left (b^{4} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} - 24 \, b x^{2} \log \left (f\right ) + 24\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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