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

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 = 0.0185339, 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.

[In]

Int[(a + b*E^(n*x))/E^(n*x),x]

[Out]

-(a/(E^(n*x)*n)) + b*x

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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.

[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 = 0.009, 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.

[In]

int((a+b*exp(n*x))/exp(n*x),x)

[Out]

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

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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.

[In]

integrate((a+b*exp(n*x))/exp(n*x),x, algorithm="maxima")

[Out]

b*x - a*e^(-n*x)/n

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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.

[In]

integrate((a+b*exp(n*x))/exp(n*x),x, algorithm="fricas")

[Out]

(b*n*x*e^(n*x) - a)*e^(-n*x)/n

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Sympy [A]  time = 0.125903, 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.

[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 [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.

[In]

integrate((a+b*exp(n*x))/exp(n*x),x, algorithm="giac")

[Out]

b*x - a*e^(-n*x)/n