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 = 0.0671766, 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 \[ -e^{-x} x^4-4 e^{-x} x^3-12 e^{-x} x^2-24 e^{-x} x-24 e^{-x} \]

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

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Rubi in Sympy [A]  time = 4.02553, size = 36, normalized size = 0.78 \[ - x^{4} e^{- x} - 4 x^{3} e^{- x} - 12 x^{2} e^{- x} - 24 x e^{- x} - 24 e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/exp(x),x)

[Out]

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

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Mathematica [A]  time = 0.00573922, 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 = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{{x}^{4}+4\,{x}^{3}+12\,{x}^{2}+24\,x+24}{{{\rm e}^{x}}}} \]

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 = 1.32588, size = 32, normalized size = 0.7 \[ -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(-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 = 0.23054, size = 32, normalized size = 0.7 \[ -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(-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 = 0.074717, size = 22, normalized size = 0.48 \[ \left (- x^{4} - 4 x^{3} - 12 x^{2} - 24 x - 24\right ) e^{- x} \]

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/XCAS [A]  time = 0.199475, size = 32, normalized size = 0.7 \[ -{\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(-x),x, algorithm="giac")

[Out]

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