Optimal. Leaf size=27 \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.0350901, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2246, 32} \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{\left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0242155, size = 26, normalized size = 0.96 \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{{\rm e}^{dx+c}} \right ) ^{1+n}}{bd \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 = 1.55541, size = 77, normalized size = 2.85 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\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 = 1.28504, size = 35, normalized size = 1.3 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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