Optimal. Leaf size=56 \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0253398, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2218} \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{(a+b x)^n} (a+b x)^m \, dx &=-\frac{(a+b x)^{1+m} \Gamma \left (\frac{1+m}{n},-(a+b x)^n \log (f)\right ) \left (-(a+b x)^n \log (f)\right )^{-\frac{1+m}{n}}}{b n}\\ \end{align*}
Mathematica [A] time = 0.0137881, size = 56, normalized size = 1. \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.34, size = 0, normalized size = 0. \begin{align*} \int{f}^{ \left ( bx+a \right ) ^{n}} \left ( bx+a \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \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 ({\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}, x\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^{\left (a + b x\right )^{n}} \left (a + b x\right )^{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{\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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