Optimal. Leaf size=32 \[ -\frac{1}{4} e^{-4 x} x^2+\frac{5}{8} e^{-4 x} x-\frac{11 e^{-4 x}}{32} \]
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Rubi [A] time = 0.0439118, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2196, 2194, 2176} \[ -\frac{1}{4} e^{-4 x} x^2+\frac{5}{8} e^{-4 x} x-\frac{11 e^{-4 x}}{32} \]
Antiderivative was successfully verified.
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Rule 2196
Rule 2194
Rule 2176
Rubi steps
\begin{align*} \int e^{-4 x} \left (2-3 x+x^2\right ) \, dx &=\int \left (2 e^{-4 x}-3 e^{-4 x} x+e^{-4 x} x^2\right ) \, dx\\ &=2 \int e^{-4 x} \, dx-3 \int e^{-4 x} x \, dx+\int e^{-4 x} x^2 \, dx\\ &=-\frac{1}{2} e^{-4 x}+\frac{3}{4} e^{-4 x} x-\frac{1}{4} e^{-4 x} x^2+\frac{1}{2} \int e^{-4 x} x \, dx-\frac{3}{4} \int e^{-4 x} \, dx\\ &=-\frac{5}{16} e^{-4 x}+\frac{5}{8} e^{-4 x} x-\frac{1}{4} e^{-4 x} x^2+\frac{1}{8} \int e^{-4 x} \, dx\\ &=-\frac{11}{32} e^{-4 x}+\frac{5}{8} e^{-4 x} x-\frac{1}{4} e^{-4 x} x^2\\ \end{align*}
Mathematica [A] time = 0.0194691, size = 19, normalized size = 0.59 \[ -\frac{1}{32} e^{-4 x} \left (8 x^2-20 x+11\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 19, normalized size = 0.6 \begin{align*} -{\frac{8\,{x}^{2}-20\,x+11}{32\,{{\rm e}^{4\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00812, size = 46, normalized size = 1.44 \begin{align*} -\frac{1}{32} \,{\left (8 \, x^{2} + 4 \, x + 1\right )} e^{\left (-4 \, x\right )} + \frac{3}{16} \,{\left (4 \, x + 1\right )} e^{\left (-4 \, x\right )} - \frac{1}{2} \, e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.701215, size = 49, normalized size = 1.53 \begin{align*} -\frac{1}{32} \,{\left (8 \, x^{2} - 20 \, x + 11\right )} e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09161, size = 15, normalized size = 0.47 \begin{align*} \frac{\left (- 8 x^{2} + 20 x - 11\right ) e^{- 4 x}}{32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2642, size = 22, normalized size = 0.69 \begin{align*} -\frac{1}{32} \,{\left (8 \, x^{2} - 20 \, x + 11\right )} e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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