Optimal. Leaf size=46 \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
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Rubi [A] time = 0.0239831, antiderivative size = 46, 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} \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^2} x^m \, dx &=-\frac{1}{2} f^a x^{1+m} \Gamma \left (\frac{1+m}{2},-b x^2 \log (f)\right ) \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-1-m)}\\ \end{align*}
Mathematica [A] time = 0.0121115, size = 46, normalized size = 1. \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 140, normalized size = 3. \begin{align*}{\frac{{f}^{a}}{2} \left ( -b \right ) ^{-{\frac{m}{2}}-{\frac{1}{2}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{m}{2}}-{\frac{1}{2}}} \left ( 2\,{\frac{{x}^{1+m} \left ( -b \right ) ^{m/2+1/2} \left ( \ln \left ( f \right ) \right ) ^{m/2+1/2} \left ( m/2+1/2 \right ) \left ( -b{x}^{2}\ln \left ( f \right ) \right ) ^{-m/2-1/2}\Gamma \left ( m/2+1/2 \right ) }{1+m}}+2\,{\frac{{x}^{1+m} \left ( -b \right ) ^{m/2+1/2} \left ( \ln \left ( f \right ) \right ) ^{m/2+1/2} \left ( -m/2-1/2 \right ) \left ( -b{x}^{2}\ln \left ( f \right ) \right ) ^{-m/2-1/2}\Gamma \left ( m/2+1/2,-b{x}^{2}\ln \left ( f \right ) \right ) }{1+m}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22648, size = 51, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, \left (-b x^{2} \log \left (f\right )\right )^{-\frac{1}{2} \, m - \frac{1}{2}} f^{a} x^{m + 1} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -b x^{2} \log \left (f\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56654, size = 126, normalized size = 2.74 \begin{align*} \frac{e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-b \log \left (f\right )\right ) + a \log \left (f\right )\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, b \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x^{2}} x^{m}\, dx \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^{b x^{2} + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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