Optimal. Leaf size=16 \[ b x-\frac{a e^{-n x}}{n} \]
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Rubi [A] time = 0.0185339, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2248, 43} \[ b x-\frac{a e^{-n x}}{n} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int e^{-n x} \left (a+b e^{n x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x}{x^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}+\frac{b}{x}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{a e^{-n x}}{n}+b x\\ \end{align*}
Mathematica [A] time = 0.0082154, size = 16, normalized size = 1. \[ b x-\frac{a e^{-n x}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 24, normalized size = 1.5 \begin{align*}{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{n}}-{\frac{a}{n{{\rm e}^{nx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12373, size = 20, normalized size = 1.25 \begin{align*} b x - \frac{a e^{\left (-n x\right )}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47915, size = 43, normalized size = 2.69 \begin{align*} \frac{{\left (b n x e^{\left (n x\right )} - a\right )} e^{\left (-n x\right )}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.125903, size = 15, normalized size = 0.94 \begin{align*} b x + \begin{cases} - \frac{a e^{- n x}}{n} & \text{for}\: n \neq 0 \\a x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25101, size = 20, normalized size = 1.25 \begin{align*} b x - \frac{a e^{\left (-n x\right )}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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