Integrand size = 11, antiderivative size = 44 \[ \int e^{-x/2} x^3 \, dx=-96 e^{-x/2}-48 e^{-x/2} x-12 e^{-x/2} x^2-2 e^{-x/2} x^3 \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2207, 2225} \[ \int e^{-x/2} x^3 \, dx=-2 e^{-x/2} x^3-12 e^{-x/2} x^2-48 e^{-x/2} x-96 e^{-x/2} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = -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*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.52 \[ \int e^{-x/2} x^3 \, dx=e^{-x/2} \left (-96-48 x-12 x^2-2 x^3\right ) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 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 | \(96-4 \left (\frac {1}{2} x^{3}+3 x^{2}+12 x +24\right ) {\mathrm e}^{-\frac {x}{2}}\) | \(24\) |
parallelrisch | \(-\left (2 x^{3}+12 x^{2}+48 x +96\right ) {\mathrm e}^{-\frac {x}{2}}\) | \(24\) |
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 | \(-96 \,{\mathrm e}^{-\frac {x}{2}}-48 x \,{\mathrm e}^{-\frac {x}{2}}-12 x^{2} {\mathrm e}^{-\frac {x}{2}}-2 x^{3} {\mathrm e}^{-\frac {x}{2}}\) | \(41\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.43 \[ \int e^{-x/2} x^3 \, dx=-2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.45 \[ \int e^{-x/2} x^3 \, dx=\left (- 2 x^{3} - 12 x^{2} - 48 x - 96\right ) e^{- \frac {x}{2}} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.43 \[ \int e^{-x/2} x^3 \, dx=-2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.43 \[ \int e^{-x/2} x^3 \, dx=-2 \, {\left (x^{3} + 6 \, x^{2} + 24 \, x + 48\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{-x/2} x^3 \, dx=-16\,{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {x^3}{8}+\frac {3\,x^2}{4}+3\,x+6\right ) \]
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