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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.00 \[ -\frac {x^{m+1} \left (-x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-x^n\right )}{n} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} e^{\left (x^{n}\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} e^{\left (x^{n}\right )}\,{d x} \]

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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int x^{m} {\mathrm e}^{x^{n}}\, dx \]

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 = 0.98, size = 38, normalized size = 1.03 \[ -\frac {x^{m + 1} \Gamma \left (\frac {m + 1}{n}, -x^{n}\right )}{n \left (-x^{n}\right )^{\frac {m + 1}{n}}} \]

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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mupad [B]  time = 3.75, size = 58, normalized size = 1.57 \[ \frac {x^{m+1}\,{\mathrm {e}}^{\frac {x^n}{2}}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (x^n\right )}{{\left (x^n\right )}^{\frac {m+n+1}{2\,n}}\,\left (m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(m + 1)*exp(x^n/2)*whittakerM(1 - (m + n + 1)/(2*n), (m + n + 1)/(2*n) - 1/2, x^n))/((x^n)^((m + n + 1)/(2*
n))*(m + 1))

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sympy [C]  time = 1.38, size = 105, normalized size = 2.84 \[ \frac {m e^{- \frac {i \pi }{n}} e^{- \frac {i \pi m}{n}} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) \gamma \left (\frac {m}{n} + \frac {1}{n}, x^{n} e^{i \pi }\right )}{n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {e^{- \frac {i \pi }{n}} e^{- \frac {i \pi m}{n}} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) \gamma \left (\frac {m}{n} + \frac {1}{n}, x^{n} e^{i \pi }\right )}{n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} \]

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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