Optimal. Leaf size=52 \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]
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Rubi [A] time = 0.0263118, antiderivative size = 52, 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} \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int e^{(a+b x)^n} (a+b x)^m \, dx &=-\frac{(a+b x)^{1+m} \left (-(a+b x)^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-(a+b x)^n\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0139617, size = 52, normalized size = 1. \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.294, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \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} e^{\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} e^{\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 \left (a + b x\right )^{m} e^{\left (a + b x\right )^{n}}\, 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} e^{\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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