3.5 \(\int e^x (a+b e^x)^n \, dx\)

Optimal. Leaf size=20 \[ \frac{\left (a+b e^x\right )^{n+1}}{b (n+1)} \]

[Out]

(a + b*E^x)^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.019931, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 32} \[ \frac{\left (a+b e^x\right )^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^x*(a + b*E^x)^n,x]

[Out]

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

Mathematica [A]  time = 0.018944, size = 19, normalized size = 0.95 \[ \frac{\left (a+b e^x\right )^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*(a + b*E^x)^n,x]

[Out]

(a + b*E^x)^(1 + n)/(b + b*n)

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Maple [A]  time = 0.002, size = 20, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{{\rm e}^{x}} \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54118, size = 50, normalized size = 2.5 \begin{align*} \frac{{\left (b e^{x} + a\right )}{\left (b e^{x} + a\right )}^{n}}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^x + a)*(b*e^x + a)^n/(b*n + b)

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Sympy [A]  time = 1.26341, size = 56, normalized size = 2.8 \begin{align*} \begin{cases} \frac{e^{x}}{a} & \text{for}\: b = 0 \wedge n = -1 \\a^{n} e^{x} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b} & \text{for}\: n = -1 \\\frac{a \left (a + b e^{x}\right )^{n}}{b n + b} + \frac{b \left (a + b e^{x}\right )^{n} e^{x}}{b n + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((exp(x)/a, Eq(b, 0) & Eq(n, -1)), (a**n*exp(x), Eq(b, 0)), (log(a/b + exp(x))/b, Eq(n, -1)), (a*(a +
b*exp(x))**n/(b*n + b) + b*(a + b*exp(x))**n*exp(x)/(b*n + b), True))

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Giac [A]  time = 1.30517, size = 26, normalized size = 1.3 \begin{align*} \frac{{\left (b e^{x} + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^x + a)^(n + 1)/(b*(n + 1))