### 3.362 $$\int e^{-x} x^4 \, dx$$

Optimal. Leaf size=46 $-e^{-x} x^4-4 e^{-x} x^3-12 e^{-x} x^2-24 e^{-x} x-24 e^{-x}$

[Out]

-24/E^x - (24*x)/E^x - (12*x^2)/E^x - (4*x^3)/E^x - x^4/E^x

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Rubi [A]  time = 0.0403911, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2176, 2194} $-e^{-x} x^4-4 e^{-x} x^3-12 e^{-x} x^2-24 e^{-x} x-24 e^{-x}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^4/E^x,x]

[Out]

-24/E^x - (24*x)/E^x - (12*x^2)/E^x - (4*x^3)/E^x - x^4/E^x

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !\$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{-x} x^4 \, dx &=-e^{-x} x^4+4 \int e^{-x} x^3 \, dx\\ &=-4 e^{-x} x^3-e^{-x} x^4+12 \int e^{-x} x^2 \, dx\\ &=-12 e^{-x} x^2-4 e^{-x} x^3-e^{-x} x^4+24 \int e^{-x} x \, dx\\ &=-24 e^{-x} x-12 e^{-x} x^2-4 e^{-x} x^3-e^{-x} x^4+24 \int e^{-x} \, dx\\ &=-24 e^{-x}-24 e^{-x} x-12 e^{-x} x^2-4 e^{-x} x^3-e^{-x} x^4\\ \end{align*}

Mathematica [A]  time = 0.0072785, size = 26, normalized size = 0.57 $e^{-x} \left (-x^4-4 x^3-12 x^2-24 x-24\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^4/E^x,x]

[Out]

(-24 - 24*x - 12*x^2 - 4*x^3 - x^4)/E^x

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Maple [A]  time = 0.002, size = 25, normalized size = 0.5 \begin{align*} -{\frac{{x}^{4}+4\,{x}^{3}+12\,{x}^{2}+24\,x+24}{{{\rm e}^{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/exp(x),x)

[Out]

-(x^4+4*x^3+12*x^2+24*x+24)/exp(x)

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Maxima [A]  time = 0.924995, size = 32, normalized size = 0.7 \begin{align*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/exp(x),x, algorithm="maxima")

[Out]

-(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x)

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Fricas [A]  time = 1.96542, size = 59, normalized size = 1.28 \begin{align*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/exp(x),x, algorithm="fricas")

[Out]

-(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x)

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Sympy [A]  time = 0.082029, size = 22, normalized size = 0.48 \begin{align*} \left (- x^{4} - 4 x^{3} - 12 x^{2} - 24 x - 24\right ) e^{- x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/exp(x),x)

[Out]

(-x**4 - 4*x**3 - 12*x**2 - 24*x - 24)*exp(-x)

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Giac [A]  time = 1.05508, size = 32, normalized size = 0.7 \begin{align*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/exp(x),x, algorithm="giac")

[Out]

-(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x)