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 = 0.0392965, antiderivative size = 52, 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} \[ -\frac{1}{2} e^{8-2 x} x^3-\frac{3}{4} e^{8-2 x} x^2-\frac{3}{4} e^{8-2 x} x-\frac{3}{8} e^{8-2 x} \]
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*}
Mathematica [A] time = 0.0097179, size = 26, normalized size = 0.5 \[ -\frac{1}{8} e^{8-2 x} \left (4 x^3+6 x^2+6 x+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 24, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 4\,{x}^{3}+6\,{x}^{2}+6\,x+3 \right ){{\rm e}^{8-2\,x}}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962147, size = 41, normalized size = 0.79 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.829967, size = 61, normalized size = 1.17 \begin{align*} -\frac{1}{8} \,{\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.093912, size = 24, normalized size = 0.46 \begin{align*} \frac{\left (- 4 x^{3} - 6 x^{2} - 6 x - 3\right ) e^{8 - 2 x}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27472, size = 31, normalized size = 0.6 \begin{align*} -\frac{1}{8} \,{\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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