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.
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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*}
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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