3.39 \(\int e^{-x} x^2 \, dx\)

Optimal. Leaf size=26 \[ -e^{-x} x^2-2 e^{-x} x-2 e^{-x} \]

[Out]

-2/E^x - (2*x)/E^x - x^2/E^x

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Rubi [A]  time = 0.0321554, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -e^{-x} x^2-2 e^{-x} x-2 e^{-x} \]

Antiderivative was successfully verified.

[In]  Int[x^2/E^x,x]

[Out]

-2/E^x - (2*x)/E^x - x^2/E^x

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Rubi in Sympy [A]  time = 1.95296, size = 19, normalized size = 0.73 \[ - x^{2} e^{- x} - 2 x e^{- x} - 2 e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/exp(x),x)

[Out]

-x**2*exp(-x) - 2*x*exp(-x) - 2*exp(-x)

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Mathematica [A]  time = 0.00363885, size = 16, normalized size = 0.62 \[ e^{-x} \left (-x^2-2 x-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/E^x,x]

[Out]

(-2 - 2*x - x^2)/E^x

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Maple [A]  time = 0.003, size = 15, normalized size = 0.6 \[ -{\frac{{x}^{2}+2\,x+2}{{{\rm e}^{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/exp(x),x)

[Out]

-(x^2+2*x+2)/exp(x)

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Maxima [A]  time = 1.35939, size = 19, normalized size = 0.73 \[ -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(-x),x, algorithm="maxima")

[Out]

-(x^2 + 2*x + 2)*e^(-x)

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Fricas [A]  time = 0.242986, size = 19, normalized size = 0.73 \[ -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(-x),x, algorithm="fricas")

[Out]

-(x^2 + 2*x + 2)*e^(-x)

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Sympy [A]  time = 0.069028, size = 12, normalized size = 0.46 \[ \left (- x^{2} - 2 x - 2\right ) e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/exp(x),x)

[Out]

(-x**2 - 2*x - 2)*exp(-x)

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GIAC/XCAS [A]  time = 0.218072, size = 19, normalized size = 0.73 \[ -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(-x),x, algorithm="giac")

[Out]

-(x^2 + 2*x + 2)*e^(-x)