Optimal. Leaf size=52 \[ -\frac {1}{2} e^{8-2 x} x^3-\frac {3}{4} e^{8-2 x} x^2-\frac {3}{4} e^{8-2 x} x-\frac {3}{8} e^{8-2 x} \]
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Rubi [A] time = 0.04, antiderivative size = 52, 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} \[ -\frac {1}{2} e^{8-2 x} x^3-\frac {3}{4} e^{8-2 x} x^2-\frac {3}{4} e^{8-2 x} x-\frac {3}{8} e^{8-2 x} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{8-2 x} x^3 \, dx &=-\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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.50 \[ -\frac {1}{8} e^{8-2 x} \left (4 x^3+6 x^2+6 x+3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 23, normalized size = 0.44 \[ -\frac {1}{8} \, {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 23, normalized size = 0.44 \[ -\frac {1}{8} \, {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.46 \[ -\frac {\left (4 x^{3}+6 x^{2}+6 x +3\right ) {\mathrm e}^{-2 x +8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 30, normalized size = 0.58 \[ -\frac {1}{8} \, {\left (4 \, x^{3} e^{8} + 6 \, x^{2} e^{8} + 6 \, x e^{8} + 3 \, e^{8}\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 23, normalized size = 0.44 \[ -{\mathrm {e}}^{8-2\,x}\,\left (\frac {x^3}{2}+\frac {3\,x^2}{4}+\frac {3\,x}{4}+\frac {3}{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 24, normalized size = 0.46 \[ \frac {\left (- 4 x^{3} - 6 x^{2} - 6 x - 3\right ) e^{8 - 2 x}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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