3.769 \(\int e^{x^n} x^m \, dx\)

Optimal. Leaf size=37 \[ -\frac{x^{m+1} \left (-x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-x^n\right )}{n} \]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -x^n])/(n*(-x^n)^((1 + m)/n)))

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Rubi [A]  time = 0.0169732, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2218} \[ -\frac{x^{m+1} \left (-x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[E^x^n*x^m,x]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -x^n])/(n*(-x^n)^((1 + m)/n)))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{x^n} x^m \, dx &=-\frac{x^{1+m} \left (-x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-x^n\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.007955, size = 37, normalized size = 1. \[ -\frac{x^{m+1} \left (-x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^n*x^m,x]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -x^n])/(n*(-x^n)^((1 + m)/n)))

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{x}^{n}}}{x}^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^n)*x^m,x)

[Out]

int(exp(x^n)*x^m,x)

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Maxima [A]  time = 1.04909, size = 51, normalized size = 1.38 \begin{align*} -\frac{x^{m + 1} \Gamma \left (\frac{m + 1}{n}, -x^{n}\right )}{n \left (-x^{n}\right )^{\frac{m + 1}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^n)*x^m,x, algorithm="maxima")

[Out]

-x^(m + 1)*gamma((m + 1)/n, -x^n)/(n*(-x^n)^((m + 1)/n))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} e^{\left (x^{n}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^n)*x^m,x, algorithm="fricas")

[Out]

integral(x^m*e^(x^n), x)

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Sympy [C]  time = 1.87454, size = 105, normalized size = 2.84 \begin{align*} \frac{m e^{- \frac{i \pi }{n}} e^{- \frac{i \pi m}{n}} \Gamma \left (\frac{m}{n} + \frac{1}{n}\right ) \gamma \left (\frac{m}{n} + \frac{1}{n}, x^{n} e^{i \pi }\right )}{n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} + \frac{e^{- \frac{i \pi }{n}} e^{- \frac{i \pi m}{n}} \Gamma \left (\frac{m}{n} + \frac{1}{n}\right ) \gamma \left (\frac{m}{n} + \frac{1}{n}, x^{n} e^{i \pi }\right )}{n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**n)*x**m,x)

[Out]

m*exp(-I*pi/n)*exp(-I*pi*m/n)*gamma(m/n + 1/n)*lowergamma(m/n + 1/n, x**n*exp_polar(I*pi))/(n**2*gamma(m/n + 1
+ 1/n)) + exp(-I*pi/n)*exp(-I*pi*m/n)*gamma(m/n + 1/n)*lowergamma(m/n + 1/n, x**n*exp_polar(I*pi))/(n**2*gamm
a(m/n + 1 + 1/n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} e^{\left (x^{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^n)*x^m,x, algorithm="giac")

[Out]

integrate(x^m*e^(x^n), x)