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

Optimal. Leaf size=27 \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0350901, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2246, 32} \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0242155, size = 26, normalized size = 0.96 \[ \frac{\left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{{\rm e}^{dx+c}} \right ) ^{1+n}}{bd \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*exp(d*x+c))^(1+n)/b/d/(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+c)*(a+b*exp(d*x+c))^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55541, size = 77, normalized size = 2.85 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\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+c)*(a+b*exp(d*x+c))^n,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.28504, size = 35, normalized size = 1.3 \begin{align*} \frac{{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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