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

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

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Rubi [A]  time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \[ -2 e^{-x/2} x^3-12 e^{-x/2} x^2-48 e^{-x/2} x-96 e^{-x/2} \]

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)

Rule 2176

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] &&  !$UseGamma === True

Rule 2194

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/2} x^3 \, dx &=-2 e^{-x/2} x^3+6 \int e^{-x/2} x^2 \, dx\\ &=-12 e^{-x/2} x^2-2 e^{-x/2} x^3+24 \int e^{-x/2} x \, dx\\ &=-48 e^{-x/2} x-12 e^{-x/2} x^2-2 e^{-x/2} x^3+48 \int e^{-x/2} \, dx\\ &=-96 e^{-x/2}-48 e^{-x/2} x-12 e^{-x/2} x^2-2 e^{-x/2} x^3\\ \end {align*}

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Mathematica [A]  time = 0.01, 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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-x/2} x^3 \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[x^3/E^(x/2),x]

[Out]

Could not integrate

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fricas [A]  time = 1.27, size = 19, normalized size = 0.43 \[ -2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(1/2*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.60, size = 19, normalized size = 0.43 \[ -2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(1/2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 21, normalized size = 0.48

method	result	size
risch	\(\left (-2 x^{3}-12 x^{2}-48 x -96\right ) {\mathrm e}^{-\frac {x}{2}}\)	\(21\)
gosper	\(-2 \left (x^{3}+6 x^{2}+24 x +48\right ) {\mathrm e}^{-\frac {x}{2}}\)	\(22\)
norman	\(\left (-2 x^{3}-12 x^{2}-48 x -96\right ) {\mathrm e}^{-\frac {x}{2}}\)	\(23\)
meijerg	\(96-4 \left (\frac {1}{2} x^{3}+3 x^{2}+12 x +24\right ) {\mathrm e}^{-\frac {x}{2}}\)	\(24\)
derivativedivides	\(-96 \,{\mathrm e}^{-\frac {x}{2}}-48 x \,{\mathrm e}^{-\frac {x}{2}}-12 x^{2} {\mathrm e}^{-\frac {x}{2}}-2 x^{3} {\mathrm e}^{-\frac {x}{2}}\)	\(41\)
default	\(-96 \,{\mathrm e}^{-\frac {x}{2}}-48 x \,{\mathrm e}^{-\frac {x}{2}}-12 x^{2} {\mathrm e}^{-\frac {x}{2}}-2 x^{3} {\mathrm e}^{-\frac {x}{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/exp(1/2*x),x,method=_RETURNVERBOSE)

[Out]

(-2*x^3-12*x^2-48*x-96)*exp(-1/2*x)

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maxima [A]  time = 0.51, size = 19, normalized size = 0.43 \[ -2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(1/2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 21, normalized size = 0.48 \[ -16\,{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {x^3}{8}+\frac {3\,x^2}{4}+3\,x+6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(-x/2),x)

[Out]

-16*exp(-x/2)*(3*x + (3*x^2)/4 + x^3/8 + 6)

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sympy [A]  time = 0.09, 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)

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