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/exp(n*x)/n+b*x

________________________________________________________________________________________

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.

[In]

Int[(a + b*E^(n*x))/E^(n*x),x]

[Out]

-(a/(E^(n*x)*n)) + b*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])

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]

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 = 0.01, size = 16, normalized size = 1.00 \[ b x-\frac {a e^{-n x}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(n*x))/E^(n*x),x]

[Out]

-(a/(E^(n*x)*n)) + b*x

________________________________________________________________________________________

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.

[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

________________________________________________________________________________________

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.

[In]

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

[Out]

b*x - a*e^(-n*x)/n

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

b*x - a*e^(-n*x)/n

________________________________________________________________________________________

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.

[In]

int(exp(-n*x)*(a + b*exp(n*x)),x)

[Out]

b*x - (a*exp(-n*x))/n

________________________________________________________________________________________

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.

[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))

________________________________________________________________________________________