3.15 \(\int e^{n x} (a+b (e^x)^n)^p \, dx\)

Optimal. Leaf size=37 \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0589087, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2247, 2246, 32} \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*x)*(a + b*(E^x)^n)^p,x]

[Out]

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

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

Mathematica [A]  time = 0.0424461, size = 36, normalized size = 0.97 \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.015, size = 52, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{nx}}{{\rm e}^{p\ln \left ( a+b{{\rm e}^{nx}} \right ) }}}{n \left ( 1+p \right ) }}+{\frac{a{{\rm e}^{p\ln \left ( a+b{{\rm e}^{nx}} \right ) }}}{bn \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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(x)^n)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57852, size = 66, normalized size = 1.78 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}{\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^(n*x) + a)*(b*e^(n*x) + a)^p/(b*n*p + b*n)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \left (e^{x}\right )^{n}\right )^{p} e^{n x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*exp(x)**n)**p*exp(n*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.26427, size = 32, normalized size = 0.86 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^(n*x) + a)^(p + 1)/(b*n*(p + 1))