3.527 \(\int e^{n x} (a+b e^{n x})^{r/s} \, dx\)

Optimal. Leaf size=30 \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}}}{b n (r+s)} \]

[Out]

((a + b*E^(n*x))^((r + s)/s)*s)/(b*n*(r + s))

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Rubi [A]  time = 0.0384192, antiderivative size = 30, 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} \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}}}{b n (r+s)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*x)*(a + b*E^(n*x))^(r/s),x]

[Out]

((a + b*E^(n*x))^((r + s)/s)*s)/(b*n*(r + s))

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

Mathematica [A]  time = 0.0381094, size = 30, normalized size = 1. \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r}{s}+1}}{b n r+b n s} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*x)*(a + b*E^(n*x))^(r/s),x]

[Out]

((a + b*E^(n*x))^(1 + r/s)*s)/(b*n*r + b*n*s)

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Maple [A]  time = 0.003, size = 33, normalized size = 1.1 \begin{align*}{\frac{1}{nb} \left ( a+b{{\rm e}^{nx}} \right ) ^{{\frac{r}{s}}+1} \left ({\frac{r}{s}}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(a+b*exp(n*x))^(r/s+1)/b/(r/s+1)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81068, size = 80, normalized size = 2.67 \begin{align*} \frac{{\left (b s e^{\left (n x\right )} + a s\right )}{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}}{b n r + b n s} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*s*e^(n*x) + a*s)*(b*e^(n*x) + a)^(r/s)/(b*n*r + b*n*s)

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Sympy [A]  time = 1.20665, size = 94, normalized size = 3.13 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge r = - s \\\frac{a^{\frac{r}{s}} e^{n x}}{n} & \text{for}\: b = 0 \\x \left (a + b\right )^{\frac{r}{s}} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + e^{n x} \right )}}{b n} & \text{for}\: r = - s \\\frac{a s \left (a + b e^{n x}\right )^{\frac{r}{s}}}{b n r + b n s} + \frac{b s \left (a + b e^{n x}\right )^{\frac{r}{s}} e^{n x}}{b n r + b n s} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*x)*(a+b*exp(n*x))**(r/s),x)

[Out]

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

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Giac [A]  time = 1.16899, size = 43, normalized size = 1.43 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s} + 1}}{b n{\left (\frac{r}{s} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^(n*x) + a)^(r/s + 1)/(b*n*(r/s + 1))