Optimal. Leaf size=46 \[ -\frac {1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b x^2 \log (f)\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ -\frac {1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b x^2 \log (f)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 40, normalized size = 0.87 \[ \frac {e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (-b \log \relax (f)\right ) + a \log \relax (f)\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2} \log \relax (f)\right )}{2 \, b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x^{2} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 140, normalized size = 3.04 \[ \frac {\left (\frac {2 \left (\frac {m}{2}+\frac {1}{2}\right ) x^{m +1} \left (-b \right )^{\frac {m}{2}+\frac {1}{2}} \left (-b \,x^{2} \ln \relax (f )\right )^{-\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{\frac {m}{2}+\frac {1}{2}} \Gamma \left (\frac {m}{2}+\frac {1}{2}\right )}{m +1}+\frac {2 \left (-\frac {m}{2}-\frac {1}{2}\right ) x^{m +1} \left (-b \right )^{\frac {m}{2}+\frac {1}{2}} \left (-b \,x^{2} \ln \relax (f )\right )^{-\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{\frac {m}{2}+\frac {1}{2}} \Gamma \left (\frac {m}{2}+\frac {1}{2}, -b \,x^{2} \ln \relax (f )\right )}{m +1}\right ) f^{a} \left (-b \right )^{-\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{-\frac {m}{2}-\frac {1}{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 38, normalized size = 0.83 \[ -\frac {1}{2} \, \left (-b x^{2} \log \relax (f)\right )^{-\frac {1}{2} \, m - \frac {1}{2}} f^{a} x^{m + 1} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2} \log \relax (f)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 49, normalized size = 1.07 \[ \frac {f^a\,x^{m+1}\,\left (\Gamma \left (\frac {m}{2}+\frac {1}{2}\right )-\Gamma \left (\frac {m}{2}+\frac {1}{2},-b\,x^2\,\ln \relax (f)\right )\right )}{2\,{\left (-b\,x^2\,\ln \relax (f)\right )}^{\frac {m}{2}+\frac {1}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x^{2}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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