Optimal. Leaf size=52 \[ -\frac {(a+b x)^{1+m} \left (-(a+b x)^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-(a+b x)^n\right )}{b n} \]
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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 = {2250}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int e^{a+c+b x^n+d x^n} \, dx &=\int e^{a+c+(b+d) x^n} \, dx\\ &=-\frac {e^{a+c} x \left (-(b+d) x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-(b+d) x^n\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 52, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\left (b x +a \right )^{n}} \left (b x +a \right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b x\right )^{m} e^{\left (a + b x\right )^{n}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.89, size = 77, normalized size = 1.48 \begin {gather*} \frac {{\mathrm {e}}^{\frac {{\left (a+b\,x\right )}^n}{2}}\,{\left (a+b\,x\right )}^{m+1}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left ({\left (a+b\,x\right )}^n\right )}{b\,{\left ({\left (a+b\,x\right )}^n\right )}^{\frac {m+n+1}{2\,n}}\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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