3.27 \(\int e^{-n x} \left (a+b e^{n x}\right ) \, dx\)

Optimal. Leaf size=16 \[ b x-\frac{a e^{-n x}}{n} \]

[Out]

-(a/(E^(n*x)*n)) + b*x

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Rubi [A]  time = 0.0350138, 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 \[ b x-\frac{a e^{-n x}}{n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(n*x))/E^(n*x),x]

[Out]

-(a/(E^(n*x)*n)) + b*x

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Rubi in Sympy [A]  time = 6.06691, size = 17, normalized size = 1.06 \[ - \frac{a e^{- n x}}{n} + \frac{b \log{\left (e^{n x} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*exp(n*x))/exp(n*x),x)

[Out]

-a*exp(-n*x)/n + b*log(exp(n*x))/n

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Mathematica [A]  time = 0.0101745, size = 16, normalized size = 1. \[ b x-\frac{a e^{-n x}}{n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^(n*x))/E^(n*x),x]

[Out]

-(a/(E^(n*x)*n)) + b*x

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Maple [A]  time = 0.013, size = 24, normalized size = 1.5 \[ -{\frac{a}{n{{\rm e}^{nx}}}}+{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*exp(n*x))/exp(n*x),x)

[Out]

-a/exp(n*x)/n+1/n*b*ln(exp(n*x))

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Maxima [A]  time = 0.780423, size = 20, normalized size = 1.25 \[ b x - \frac{a e^{\left (-n x\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)*e^(-n*x),x, algorithm="maxima")

[Out]

b*x - a*e^(-n*x)/n

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Fricas [A]  time = 0.277733, size = 28, normalized size = 1.75 \[ \frac{{\left (b n x e^{\left (n x\right )} - a\right )} e^{\left (-n x\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)*e^(-n*x),x, algorithm="fricas")

[Out]

(b*n*x*e^(n*x) - a)*e^(-n*x)/n

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Sympy [A]  time = 0.18602, 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.

[In]  integrate((a+b*exp(n*x))/exp(n*x),x)

[Out]

b*x + Piecewise((-a*exp(-n*x)/n, Ne(n, 0)), (a*x, True))

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GIAC/XCAS [A]  time = 0.244053, size = 20, normalized size = 1.25 \[ b x - \frac{a e^{\left (-n x\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)*e^(-n*x),x, algorithm="giac")

[Out]

b*x - a*e^(-n*x)/n