Integrand size = 11, antiderivative size = 52 \[ \int e^{8-2 x} x^3 \, dx=-\frac {3}{8} e^{8-2 x}-\frac {3}{4} e^{8-2 x} x-\frac {3}{4} e^{8-2 x} x^2-\frac {1}{2} e^{8-2 x} x^3 \]
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Time = 0.03 (sec) , antiderivative size = 52, 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} \[ \int e^{8-2 x} x^3 \, dx=-\frac {1}{2} e^{8-2 x} x^3-\frac {3}{4} e^{8-2 x} x^2-\frac {3}{4} e^{8-2 x} x-\frac {3}{8} e^{8-2 x} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} e^{8-2 x} x^3+\frac {3}{2} \int e^{8-2 x} x^2 \, dx \\ & = -\frac {3}{4} e^{8-2 x} x^2-\frac {1}{2} e^{8-2 x} x^3+\frac {3}{2} \int e^{8-2 x} x \, dx \\ & = -\frac {3}{4} e^{8-2 x} x-\frac {3}{4} e^{8-2 x} x^2-\frac {1}{2} e^{8-2 x} x^3+\frac {3}{4} \int e^{8-2 x} \, dx \\ & = -\frac {3}{8} e^{8-2 x}-\frac {3}{4} e^{8-2 x} x-\frac {3}{4} e^{8-2 x} x^2-\frac {1}{2} e^{8-2 x} x^3 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.50 \[ \int e^{8-2 x} x^3 \, dx=-\frac {1}{8} e^{8-2 x} \left (3+6 x+6 x^2+4 x^3\right ) \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44
method | result | size |
risch | \(\left (-\frac {1}{2} x^{3}-\frac {3}{4} x^{2}-\frac {3}{4} x -\frac {3}{8}\right ) {\mathrm e}^{8-2 x}\) | \(23\) |
gosper | \(-\frac {\left (4 x^{3}+6 x^{2}+6 x +3\right ) {\mathrm e}^{8-2 x}}{8}\) | \(24\) |
meijerg | \(\frac {{\mathrm e}^{8} \left (6-\frac {\left (32 x^{3}+48 x^{2}+48 x +24\right ) {\mathrm e}^{-2 x}}{4}\right )}{16}\) | \(28\) |
norman | \(-\frac {3 \,{\mathrm e}^{8-2 x}}{8}-\frac {3 \,{\mathrm e}^{8-2 x} x}{4}-\frac {3 \,{\mathrm e}^{8-2 x} x^{2}}{4}-\frac {{\mathrm e}^{8-2 x} x^{3}}{2}\) | \(41\) |
parallelrisch | \(-\frac {3 \,{\mathrm e}^{8-2 x}}{8}-\frac {3 \,{\mathrm e}^{8-2 x} x}{4}-\frac {3 \,{\mathrm e}^{8-2 x} x^{2}}{4}-\frac {{\mathrm e}^{8-2 x} x^{3}}{2}\) | \(41\) |
parts | \(-\frac {{\mathrm e}^{8-2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )^{2}}{16}+\frac {27 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )}{8}-\frac {123 \,{\mathrm e}^{8-2 x}}{8}\) | \(49\) |
derivativedivides | \(\frac {123 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )}{8}-\frac {379 \,{\mathrm e}^{8-2 x}}{8}-\frac {27 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )^{2}}{16}+\frac {{\mathrm e}^{8-2 x} \left (8-2 x \right )^{3}}{16}\) | \(53\) |
default | \(\frac {123 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )}{8}-\frac {379 \,{\mathrm e}^{8-2 x}}{8}-\frac {27 \,{\mathrm e}^{8-2 x} \left (8-2 x \right )^{2}}{16}+\frac {{\mathrm e}^{8-2 x} \left (8-2 x \right )^{3}}{16}\) | \(53\) |
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44 \[ \int e^{8-2 x} x^3 \, dx=-\frac {1}{8} \, {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.46 \[ \int e^{8-2 x} x^3 \, dx=\frac {\left (- 4 x^{3} - 6 x^{2} - 6 x - 3\right ) e^{8 - 2 x}}{8} \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.58 \[ \int e^{8-2 x} x^3 \, dx=-\frac {1}{8} \, {\left (4 \, x^{3} e^{8} + 6 \, x^{2} e^{8} + 6 \, x e^{8} + 3 \, e^{8}\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44 \[ \int e^{8-2 x} x^3 \, dx=-\frac {1}{8} \, {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \]
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Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44 \[ \int e^{8-2 x} x^3 \, dx=-{\mathrm {e}}^{8-2\,x}\,\left (\frac {x^3}{2}+\frac {3\,x^2}{4}+\frac {3\,x}{4}+\frac {3}{8}\right ) \]
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