Optimal. Leaf size=16 \[ -\frac {a e^{-n x}}{n}+b x \]
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Rubi [A]
time = 0.01, 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 = {2280, 45}
\begin {gather*} b x-\frac {a e^{-n x}}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2280
Rubi steps
\begin {align*} \int e^{-n x} \left (a+b e^{n x}\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b x}{x^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\text {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 \begin {gather*} -\frac {a e^{-n x}}{n}+b x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 22, normalized size = 1.38
method | result | size |
risch | \(-\frac {a \,{\mathrm e}^{-n x}}{n}+b x\) | \(16\) |
derivativedivides | \(\frac {-a \,{\mathrm e}^{-n x}+b \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(22\) |
default | \(\frac {-a \,{\mathrm e}^{-n x}+b \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(22\) |
norman | \(\left (b x \,{\mathrm e}^{n x}-\frac {a}{n}\right ) {\mathrm e}^{-n x}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 0.94 \begin {gather*} b x - \frac {a e^{\left (-n x\right )}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 1.31 \begin {gather*} \frac {{\left (b n x e^{\left (n x\right )} - a\right )} e^{\left (-n x\right )}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.94 \begin {gather*} b x + \begin {cases} - \frac {a e^{- n x}}{n} & \text {for}\: n \neq 0 \\a x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.32, size = 15, normalized size = 0.94 \begin {gather*} b x - \frac {a e^{\left (-n x\right )}}{n} \end {gather*}
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 \begin {gather*} b\,x-\frac {a\,{\mathrm {e}}^{-n\,x}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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