3.535 \(\int e^{-x/2} x^3 \, dx\)

Optimal. Leaf size=44 \[ -2 e^{-x/2} x^3-12 e^{-x/2} x^2-48 e^{-x/2} x-96 e^{-x/2} \]

[Out]

-96/E^(x/2) - (48*x)/E^(x/2) - (12*x^2)/E^(x/2) - (2*x^3)/E^(x/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0544157, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -2 e^{-x/2} x^3-12 e^{-x/2} x^2-48 e^{-x/2} x-96 e^{-x/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/E^(x/2),x]

[Out]

-96/E^(x/2) - (48*x)/E^(x/2) - (12*x^2)/E^(x/2) - (2*x^3)/E^(x/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.04113, size = 36, normalized size = 0.82 \[ - 2 x^{3} e^{- \frac{x}{2}} - 12 x^{2} e^{- \frac{x}{2}} - 48 x e^{- \frac{x}{2}} - 96 e^{- \frac{x}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/exp(1/2*x),x)

[Out]

-2*x**3*exp(-x/2) - 12*x**2*exp(-x/2) - 48*x*exp(-x/2) - 96*exp(-x/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00522756, size = 23, normalized size = 0.52 \[ e^{-x/2} \left (-2 x^3-12 x^2-48 x-96\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/E^(x/2),x]

[Out]

(-96 - 48*x - 12*x^2 - 2*x^3)/E^(x/2)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \[ -2\,{\frac{{x}^{3}+6\,{x}^{2}+24\,x+48}{{{\rm e}^{x/2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/exp(1/2*x),x)

[Out]

-2*(x^3+6*x^2+24*x+48)/exp(1/2*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.37629, size = 26, normalized size = 0.59 \[ -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-1/2*x),x, algorithm="maxima")

[Out]

-2*(x^3 + 6*x^2 + 24*x + 48)*e^(-1/2*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.214349, size = 26, normalized size = 0.59 \[ -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-1/2*x),x, algorithm="fricas")

[Out]

-2*(x^3 + 6*x^2 + 24*x + 48)*e^(-1/2*x)

_______________________________________________________________________________________

Sympy [A]  time = 0.075999, size = 20, normalized size = 0.45 \[ \left (- 2 x^{3} - 12 x^{2} - 48 x - 96\right ) e^{- \frac{x}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/exp(1/2*x),x)

[Out]

(-2*x**3 - 12*x**2 - 48*x - 96)*exp(-x/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21273, size = 26, normalized size = 0.59 \[ -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-1/2*x),x, algorithm="giac")

[Out]

-2*(x^3 + 6*x^2 + 24*x + 48)*e^(-1/2*x)