Optimal. Leaf size=46 \[ -\frac {f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n \log (f)\right )}{n} \]
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Rubi [A] time = 0.03, 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} \[ -\frac {f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b \log (f) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+b x^n} x^m \, dx &=-\frac {f^a x^{1+m} \Gamma \left (\frac {1+m}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-\frac {1+m}{n}}}{n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ -\frac {f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a} x^{m}, x\right ) \]
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 \[ \int f^{b x^{n} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int f^{b \,x^{n}+a} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 47, normalized size = 1.02 \[ -\frac {f^{a} x^{m + 1} \Gamma \left (\frac {m + 1}{n}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\frac {m + 1}{n}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.76, size = 79, normalized size = 1.72 \[ \frac {f^a\,f^{b\,x^n}\,x^{m+1}\,{\mathrm {e}}^{-\frac {b\,x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (b\,x^n\,\ln \relax (f)\right )}{\left (m+1\right )\,{\left (b\,x^n\,\ln \relax (f)\right )}^{\frac {m+n+1}{2\,n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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