3.286 \(\int e^{-x^3} x^5 \, dx\)

Optimal. Leaf size=26 \[ -\frac{1}{3} e^{-x^3} x^3-\frac{e^{-x^3}}{3} \]

[Out]

-1/(3*E^x^3) - x^3/(3*E^x^3)

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Rubi [A]  time = 0.0266444, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2212, 2209} \[ -\frac{1}{3} e^{-x^3} x^3-\frac{e^{-x^3}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/E^x^3,x]

[Out]

-1/(3*E^x^3) - x^3/(3*E^x^3)

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

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

Rubi steps

\begin{align*} \int e^{-x^3} x^5 \, dx &=-\frac{1}{3} e^{-x^3} x^3+\int e^{-x^3} x^2 \, dx\\ &=-\frac{e^{-x^3}}{3}-\frac{1}{3} e^{-x^3} x^3\\ \end{align*}

Mathematica [A]  time = 0.0030772, size = 16, normalized size = 0.62 \[ -\frac{1}{3} e^{-x^3} \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/E^x^3,x]

[Out]

-(1 + x^3)/(3*E^x^3)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.5 \begin{align*} -{\frac{{x}^{3}+1}{3\,{{\rm e}^{{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/exp(x^3),x)

[Out]

-1/3*(x^3+1)/exp(x^3)

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Maxima [A]  time = 0.96886, size = 18, normalized size = 0.69 \begin{align*} -\frac{1}{3} \,{\left (x^{3} + 1\right )} e^{\left (-x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/exp(x^3),x, algorithm="maxima")

[Out]

-1/3*(x^3 + 1)*e^(-x^3)

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Fricas [A]  time = 1.92135, size = 34, normalized size = 1.31 \begin{align*} -\frac{1}{3} \,{\left (x^{3} + 1\right )} e^{\left (-x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/exp(x^3),x, algorithm="fricas")

[Out]

-1/3*(x^3 + 1)*e^(-x^3)

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Sympy [A]  time = 0.08548, size = 12, normalized size = 0.46 \begin{align*} \frac{\left (- x^{3} - 1\right ) e^{- x^{3}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/exp(x**3),x)

[Out]

(-x**3 - 1)*exp(-x**3)/3

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Giac [A]  time = 1.04592, size = 18, normalized size = 0.69 \begin{align*} -\frac{1}{3} \,{\left (x^{3} + 1\right )} e^{\left (-x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/exp(x^3),x, algorithm="giac")

[Out]

-1/3*(x^3 + 1)*e^(-x^3)