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 = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 2176
Rule 2194
Rule 2196
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.39, size = 16, normalized size = 0.50 \[ -\frac {1}{32} \, {\left (8 \, x^{2} - 20 \, x + 11\right )} e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 16, normalized size = 0.50 \[ -\frac {1}{32} \, {\left (8 \, x^{2} - 20 \, x + 11\right )} e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 0.59 \[ -\frac {\left (8 x^{2}-20 x +11\right ) {\mathrm e}^{-4 x}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 34, normalized size = 1.06 \[ -\frac {1}{32} \, {\left (8 \, x^{2} + 4 \, x + 1\right )} e^{\left (-4 \, x\right )} + \frac {3}{16} \, {\left (4 \, x + 1\right )} e^{\left (-4 \, x\right )} - \frac {1}{2} \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 0.50 \[ -\frac {{\mathrm {e}}^{-4\,x}\,\left (8\,x^2-20\,x+11\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 15, normalized size = 0.47 \[ \frac {\left (- 8 x^{2} + 20 x - 11\right ) e^{- 4 x}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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