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

Optimal. Leaf size=32 \[ -\frac{1}{2} e^{-2 x} x^2-\frac{1}{2} e^{-2 x} x-\frac{e^{-2 x}}{4} \]

[Out]

-1/(4*E^(2*x)) - x/(2*E^(2*x)) - x^2/(2*E^(2*x))

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Rubi [A]  time = 0.0300944, antiderivative size = 32, 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 \[ -\frac{1}{2} e^{-2 x} x^2-\frac{1}{2} e^{-2 x} x-\frac{e^{-2 x}}{4} \]

Antiderivative was successfully verified.

[In]  Int[x^2/E^(2*x),x]

[Out]

-1/(4*E^(2*x)) - x/(2*E^(2*x)) - x^2/(2*E^(2*x))

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Rubi in Sympy [A]  time = 1.89209, size = 27, normalized size = 0.84 \[ - \frac{x^{2} e^{- 2 x}}{2} - \frac{x e^{- 2 x}}{2} - \frac{e^{- 2 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/exp(2*x),x)

[Out]

-x**2*exp(-2*x)/2 - x*exp(-2*x)/2 - exp(-2*x)/4

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Mathematica [A]  time = 0.00410474, size = 19, normalized size = 0.59 \[ -\frac{1}{4} e^{-2 x} \left (2 x^2+2 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/E^(2*x),x]

[Out]

-(1 + 2*x + 2*x^2)/(4*E^(2*x))

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Maple [A]  time = 0.003, size = 19, normalized size = 0.6 \[ -{\frac{2\,{x}^{2}+2\,x+1}{4\,{{\rm e}^{2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/exp(2*x),x)

[Out]

-1/4*(2*x^2+2*x+1)/exp(2*x)

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Maxima [A]  time = 1.41579, size = 22, normalized size = 0.69 \[ -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*x^2 + 2*x + 1)*e^(-2*x)

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Fricas [A]  time = 0.208442, size = 22, normalized size = 0.69 \[ -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*x^2 + 2*x + 1)*e^(-2*x)

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Sympy [A]  time = 0.078853, size = 17, normalized size = 0.53 \[ \frac{\left (- 2 x^{2} - 2 x - 1\right ) e^{- 2 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/exp(2*x),x)

[Out]

(-2*x**2 - 2*x - 1)*exp(-2*x)/4

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GIAC/XCAS [A]  time = 0.237221, size = 22, normalized size = 0.69 \[ -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*x^2 + 2*x + 1)*e^(-2*x)