Optimal. Leaf size=46 \[ \frac {1}{2} f^a x^{m+1} \left (-\frac {b \log (f)}{x^2}\right )^{\frac {m+1}{2}} \Gamma \left (\frac {1}{2} (-m-1),-\frac {b \log (f)}{x^2}\right ) \]
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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} \[ \frac {1}{2} f^a x^{m+1} \left (-\frac {b \log (f)}{x^2}\right )^{\frac {m+1}{2}} \text {Gamma}\left (\frac {1}{2} (-m-1),-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^m \, dx &=\frac {1}{2} f^a x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{\frac {1+m}{2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ \frac {1}{2} f^a x^{m+1} \left (-\frac {b \log (f)}{x^2}\right )^{\frac {m+1}{2}} \Gamma \left (\frac {1}{2} (-m-1),-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{\frac {a x^{2} + b}{x^{2}}} x^{m}, x\right ) \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 169, normalized size = 3.67 \[ -\frac {\left (\frac {2 b \,x^{m -1} \left (-b \right )^{-\frac {m}{2}-\frac {1}{2}} \left (-\frac {b \ln \relax (f )}{x^{2}}\right )^{\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{-\frac {m}{2}+\frac {1}{2}} \Gamma \left (-\frac {m}{2}+\frac {1}{2}\right )}{m +1}-\frac {2 b \,x^{m -1} \left (-b \right )^{-\frac {m}{2}-\frac {1}{2}} \left (-\frac {b \ln \relax (f )}{x^{2}}\right )^{\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{-\frac {m}{2}+\frac {1}{2}} \Gamma \left (-\frac {m}{2}+\frac {1}{2}, -\frac {b \ln \relax (f )}{x^{2}}\right )}{m +1}-\frac {2 x^{m +1} \left (-b \right )^{-\frac {m}{2}-\frac {1}{2}} \ln \relax (f )^{-\frac {m}{2}-\frac {1}{2}} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{2}}}}{m +1}\right ) f^{a} \left (-b \right )^{\frac {m}{2}+\frac {1}{2}} \ln \relax (f )^{\frac {m}{2}+\frac {1}{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 38, normalized size = 0.83 \[ \frac {1}{2} \, f^{a} x^{m + 1} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {1}{2} \, m + \frac {1}{2}} \Gamma \left (-\frac {1}{2} \, m - \frac {1}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 54, normalized size = 1.17 \[ \frac {f^a\,x^{m+1}\,{\mathrm {e}}^{\frac {b\,\ln \relax (f)}{2\,x^2}}\,{\mathrm {M}}_{\frac {m}{4}+\frac {3}{4},-\frac {m}{4}-\frac {1}{4}}\left (\frac {b\,\ln \relax (f)}{x^2}\right )\,{\left (\frac {b\,\ln \relax (f)}{x^2}\right )}^{\frac {m}{4}-\frac {1}{4}}}{m+1} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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