Optimal. Leaf size=25 \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0373275, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2246, 32} \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int \left (e^x\right )^n \left (a+b \left (e^x\right )^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^p \, dx,x,\left (e^x\right )^n\right )}{n}\\ &=\frac{\left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0650999, size = 24, normalized size = 0.96 \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( a+b \left ({{\rm e}^{x}} \right ) ^{n} \right ) ^{1+p}}{bn \left ( 1+p \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 = 1.50802, size = 66, normalized size = 2.64 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}{\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.21396, size = 80, normalized size = 3.2 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac{a^{p} \left (e^{x}\right )^{n}}{n} & \text{for}\: b = 0 \\x \left (a + b\right )^{p} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + \left (e^{x}\right )^{n} \right )}}{b n} & \text{for}\: p = -1 \\\frac{a \left (a + b \left (e^{x}\right )^{n}\right )^{p}}{b n p + b n} + \frac{b \left (a + b \left (e^{x}\right )^{n}\right )^{p} \left (e^{x}\right )^{n}}{b n p + b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21202, size = 32, normalized size = 1.28 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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