3.4.62 \(\int e^{-x} x^4 \, dx\) [362]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} -e^{-x} x^4-4 e^{-x} x^3-12 e^{-x} x^2-24 e^{-x} x-24 e^{-x} \end {gather*}

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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.57 \begin {gather*} e^{-x} \left (-24-24 x-12 x^2-4 x^3-x^4\right ) \end {gather*}

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 = 0.03, size = 42, normalized size = 0.91

method	result	size
gosper	\(-\left (x^{4}+4 x^{3}+12 x^{2}+24 x +24\right ) {\mathrm e}^{-x}\)	\(25\)
norman	\(\left (-x^{4}-4 x^{3}-12 x^{2}-24 x -24\right ) {\mathrm e}^{-x}\)	\(26\)
risch	\(\left (-x^{4}-4 x^{3}-12 x^{2}-24 x -24\right ) {\mathrm e}^{-x}\)	\(26\)
meijerg	\(24-\frac {\left (5 x^{4}+20 x^{3}+60 x^{2}+120 x +120\right ) {\mathrm e}^{-x}}{5}\)	\(29\)
default	\(-24 \,{\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-12 x^{2} {\mathrm e}^{-x}-4 x^{3} {\mathrm e}^{-x}-x^{4} {\mathrm e}^{-x}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/exp(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.77, size = 24, normalized size = 0.52 \begin {gather*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end {gather*}

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 = 0.54, size = 24, normalized size = 0.52 \begin {gather*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end {gather*}

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 = 0.02, size = 22, normalized size = 0.48 \begin {gather*} \left (- x^{4} - 4 x^{3} - 12 x^{2} - 24 x - 24\right ) e^{- x} \end {gather*}

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 = 0.77, size = 24, normalized size = 0.52 \begin {gather*} -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.52 \begin {gather*} -{\mathrm {e}}^{-x}\,\left (x^4+4\,x^3+12\,x^2+24\,x+24\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*exp(-x),x)

[Out]

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

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