Optimal. Leaf size=46 \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]
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Rubi [A] time = 0.0264467, 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} \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]
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*}
Mathematica [A] time = 0.0123183, size = 46, normalized size = 1. \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{f}^{a+b{x}^{n}}{x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22754, size = 63, normalized size = 1.37 \begin{align*} -\frac{f^{a} x^{m + 1} \Gamma \left (\frac{m + 1}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac{m + 1}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{b x^{n} + a} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{n} + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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