Optimal. Leaf size=30 \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}}}{b n (r+s)} \]
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Rubi [A] time = 0.0384192, antiderivative size = 30, 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} \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}}}{b n (r+s)} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^{r/s} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\left (a+b e^{n x}\right )^{\frac{r+s}{s}} s}{b n (r+s)}\\ \end{align*}
Mathematica [A] time = 0.0381094, size = 30, normalized size = 1. \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r}{s}+1}}{b n r+b n s} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 1.1 \begin{align*}{\frac{1}{nb} \left ( a+b{{\rm e}^{nx}} \right ) ^{{\frac{r}{s}}+1} \left ({\frac{r}{s}}+1 \right ) ^{-1}} \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 = 1.81068, size = 80, normalized size = 2.67 \begin{align*} \frac{{\left (b s e^{\left (n x\right )} + a s\right )}{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}}{b n r + b n s} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20665, size = 94, normalized size = 3.13 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge r = - s \\\frac{a^{\frac{r}{s}} e^{n x}}{n} & \text{for}\: b = 0 \\x \left (a + b\right )^{\frac{r}{s}} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + e^{n x} \right )}}{b n} & \text{for}\: r = - s \\\frac{a s \left (a + b e^{n x}\right )^{\frac{r}{s}}}{b n r + b n s} + \frac{b s \left (a + b e^{n x}\right )^{\frac{r}{s}} e^{n x}}{b n r + b n s} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16899, size = 43, normalized size = 1.43 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s} + 1}}{b n{\left (\frac{r}{s} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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