3.6 \(\int e^{d x} (a+b e^{c+d x})^n \, dx\)

Optimal. Leaf size=32 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(b*d*E^c*(1 + n))

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Rubi [A]  time = 0.0696912, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2247, 2246, 32} \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(d*x)*(a + b*E^(c + d*x))^n,x]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(b*d*E^c*(1 + n))

Rule 2247

Int[((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.)*((G_)^((h_.)*((f_.) + (g_.)*(x_))))^(m_.),
x_Symbol] :> Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)
^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int e^{d x} \left (a+b e^{c+d x}\right )^n \, dx &=e^{-c} \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx\\ &=\frac{e^{-c} \operatorname{Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{e^{-c} \left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.034068, size = 31, normalized size = 0.97 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(d*x)*(a + b*E^(c + d*x))^n,x]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(E^c*(b*d + b*d*n))

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Maple [A]  time = 0.002, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{{\rm e}^{dx}}{{\rm e}^{c}} \right ) ^{1+n}}{bd{{\rm e}^{c}} \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x)*(a+b*exp(d*x+c))^n,x)

[Out]

1/d*(a+b*exp(d*x)*exp(c))^(1+n)/b/exp(c)/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x)*(a+b*exp(d*x+c))^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54729, size = 81, normalized size = 2.53 \begin{align*} \frac{{\left (b e^{\left (d x\right )} + a e^{\left (-c\right )}\right )}{\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x)*(a+b*exp(d*x+c))^n,x, algorithm="fricas")

[Out]

(b*e^(d*x) + a*e^(-c))*(b*e^(d*x + c) + a)^n/(b*d*n + b*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x)*(a+b*exp(d*x+c))**n,x)

[Out]

Timed out

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Giac [A]  time = 1.21372, size = 41, normalized size = 1.28 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1} e^{\left (-c\right )}}{b d{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x)*(a+b*exp(d*x+c))^n,x, algorithm="giac")

[Out]

(b*e^(d*x + c) + a)^(n + 1)*e^(-c)/(b*d*(n + 1))