Optimal. Leaf size=32 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.0696912, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2247, 2246, 32} \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2247
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int e^{d x} \left (a+b e^{c+d x}\right )^n \, dx &=e^{-c} \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx\\ &=\frac{e^{-c} \operatorname{Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{e^{-c} \left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.034068, size = 31, normalized size = 0.97 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{{\rm e}^{dx}}{{\rm e}^{c}} \right ) ^{1+n}}{bd{{\rm e}^{c}} \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54729, size = 81, normalized size = 2.53 \begin{align*} \frac{{\left (b e^{\left (d x\right )} + a e^{\left (-c\right )}\right )}{\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \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 [A] time = 1.21372, size = 41, normalized size = 1.28 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1} e^{\left (-c\right )}}{b d{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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