Optimal. Leaf size=32 \[ -\frac{1}{2} e^{-2 x} x^2-\frac{1}{2} e^{-2 x} x-\frac{e^{-2 x}}{4} \]
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Rubi [A] time = 0.0176593, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2176, 2194} \[ -\frac{1}{2} e^{-2 x} x^2-\frac{1}{2} e^{-2 x} x-\frac{e^{-2 x}}{4} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-2 x} x^2 \, dx &=-\frac{1}{2} e^{-2 x} x^2+\int e^{-2 x} x \, dx\\ &=-\frac{1}{2} e^{-2 x} x-\frac{1}{2} e^{-2 x} x^2+\frac{1}{2} \int e^{-2 x} \, dx\\ &=-\frac{1}{4} e^{-2 x}-\frac{1}{2} e^{-2 x} x-\frac{1}{2} e^{-2 x} x^2\\ \end{align*}
Mathematica [A] time = 0.0063322, size = 19, normalized size = 0.59 \[ -\frac{1}{4} e^{-2 x} \left (2 x^2+2 x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.6 \begin{align*} -{\frac{2\,{x}^{2}+2\,x+1}{4\,{{\rm e}^{2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96247, size = 22, normalized size = 0.69 \begin{align*} -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\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 = 0.486861, size = 45, normalized size = 1.41 \begin{align*} -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081091, size = 17, normalized size = 0.53 \begin{align*} \frac{\left (- 2 x^{2} - 2 x - 1\right ) e^{- 2 x}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08076, size = 22, normalized size = 0.69 \begin{align*} -\frac{1}{4} \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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