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 = 0.0317714, 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, 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.
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Rule 2176
Rule 2194
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*}
Mathematica [A] time = 0.0071218, size = 23, normalized size = 0.52 \[ e^{-x/2} \left (-2 x^3-12 x^2-48 x-96\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.5 \begin{align*} -2\,{\frac{{x}^{3}+6\,{x}^{2}+24\,x+48}{{{\rm e}^{x/2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937732, size = 26, normalized size = 0.59 \begin{align*} -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75255, size = 55, normalized size = 1.25 \begin{align*} -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.084769, size = 20, normalized size = 0.45 \begin{align*} \left (- 2 x^{3} - 12 x^{2} - 48 x - 96\right ) e^{- \frac{x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08907, size = 26, normalized size = 0.59 \begin{align*} -2 \,{\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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