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

Optimal. Leaf size=25 \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]

[Out]

(a + b*(E^x)^n)^(1 + p)/(b*n*(1 + p))

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Rubi [A]  time = 0.0373275, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2246, 32} \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(E^x)^n*(a + b*(E^x)^n)^p,x]

[Out]

(a + b*(E^x)^n)^(1 + p)/(b*n*(1 + p))

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

Mathematica [A]  time = 0.0650999, size = 24, normalized size = 0.96 \[ \frac{\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*(E^x)^n)^(1 + p)/(b*n + b*n*p)

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Maple [A]  time = 0.001, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( a+b \left ({{\rm e}^{x}} \right ) ^{n} \right ) ^{1+p}}{bn \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50802, size = 66, normalized size = 2.64 \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(x)^n*(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)

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Sympy [A]  time = 3.21396, size = 80, normalized size = 3.2 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac{a^{p} \left (e^{x}\right )^{n}}{n} & \text{for}\: b = 0 \\x \left (a + b\right )^{p} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + \left (e^{x}\right )^{n} \right )}}{b n} & \text{for}\: p = -1 \\\frac{a \left (a + b \left (e^{x}\right )^{n}\right )^{p}}{b n p + b n} + \frac{b \left (a + b \left (e^{x}\right )^{n}\right )^{p} \left (e^{x}\right )^{n}}{b n p + b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21202, size = 32, normalized size = 1.28 \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(x)^n*(a+b*exp(x)^n)^p,x, algorithm="giac")

[Out]

(b*e^(n*x) + a)^(p + 1)/(b*n*(p + 1))