Optimal. Leaf size=16 \[ b x-\frac {a e^{-n x}}{n} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ b x-\frac {a e^{-n x}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 21, normalized size = 1.31 \[ \frac {{\left (b n x e^{\left (n x\right )} - a\right )} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 15, normalized size = 0.94 \[ b x - \frac {a e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 1.50 \[ -\frac {a \,{\mathrm e}^{-n x}}{n}+\frac {b \ln \left ({\mathrm e}^{n x}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 15, normalized size = 0.94 \[ b x - \frac {a e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 15, normalized size = 0.94 \[ b\,x-\frac {a\,{\mathrm {e}}^{-n\,x}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.94 \[ b x + \begin {cases} - \frac {a e^{- n x}}{n} & \text {for}\: n \neq 0 \\a x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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