Optimal. Leaf size=37 \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]
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Rubi [A] time = 0.0351137, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6741, 2208} \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 2208
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*}
Mathematica [A] time = 0.01572, size = 37, normalized size = 1. \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.24, size = 241, normalized size = 6.5 \begin{align*}{\frac{{{\rm e}^{a+c}}}{n} \left ( -b-d \right ) ^{-{n}^{-1}} \left ({\frac{{n}^{2}{x}^{1-n} \left ( -b-d \right ) ^{{n}^{-1}-1} \left ( n{x}^{n} \left ( -b-d \right ) +n+1 \right ) }{ \left ( 1+n \right ) \left ( 1+2\,n \right ) } \left ({x}^{n} \left ( -b-d \right ) \right ) ^{-{\frac{1+n}{2\,n}}}{{\rm e}^{-{\frac{{x}^{n} \left ( -b-d \right ) }{2}}}}{{\sl M}_{{n}^{-1}-{\frac{1+n}{2\,n}},\,{\frac{1+n}{2\,n}}+{\frac{1}{2}}}\left ({x}^{n} \left ( -b-d \right ) \right )}}+{\frac{n{x}^{1-n} \left ( -b-d \right ) ^{{n}^{-1}-1} \left ( 1+n \right ) }{1+2\,n} \left ({x}^{n} \left ( -b-d \right ) \right ) ^{-{\frac{1+n}{2\,n}}}{{\rm e}^{-{\frac{{x}^{n} \left ( -b-d \right ) }{2}}}}{{\sl M}_{{n}^{-1}-{\frac{1+n}{2\,n}}+1,\,{\frac{1+n}{2\,n}}+{\frac{1}{2}}}\left ({x}^{n} \left ( -b-d \right ) \right )}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09025, size = 49, normalized size = 1.32 \begin{align*} -\frac{x e^{\left (a + c\right )} \Gamma \left (\frac{1}{n}, -{\left (b + d\right )} x^{n}\right )}{\left (-{\left (b + d\right )} x^{n}\right )^{\left (\frac{1}{n}\right )} 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 (e^{\left ({\left (b + d\right )} x^{n} + a + c\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*} e^{a} e^{c} \int e^{b x^{n}} e^{d x^{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 e^{\left (b x^{n} + d x^{n} + a + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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