Optimal. Leaf size=26 \[ \left (3 e^{-x}+\frac {1}{2} e^{-x} (9-x)\right ) (-1+x) \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {12, 2196, 2194, 2176} \begin {gather*} -\frac {1}{2} e^{-x} x^2+8 e^{-x} x-\frac {15 e^{-x}}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{-x} \left (31-18 x+x^2\right ) \, dx\\ &=\frac {1}{2} \int \left (31 e^{-x}-18 e^{-x} x+e^{-x} x^2\right ) \, dx\\ &=\frac {1}{2} \int e^{-x} x^2 \, dx-9 \int e^{-x} x \, dx+\frac {31}{2} \int e^{-x} \, dx\\ &=-\frac {31 e^{-x}}{2}+9 e^{-x} x-\frac {1}{2} e^{-x} x^2-9 \int e^{-x} \, dx+\int e^{-x} x \, dx\\ &=-\frac {13 e^{-x}}{2}+8 e^{-x} x-\frac {1}{2} e^{-x} x^2+\int e^{-x} \, dx\\ &=-\frac {15 e^{-x}}{2}+8 e^{-x} x-\frac {1}{2} e^{-x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 0.73 \begin {gather*} \frac {1}{2} e^{-x} \left (-15+16 x-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 14, normalized size = 0.54 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 16 \, x + 15\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 14, normalized size = 0.54 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 16 \, x + 15\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 15, normalized size = 0.58
method | result | size |
gosper | \(-\frac {\left (x^{2}-16 x +15\right ) {\mathrm e}^{-x}}{2}\) | \(15\) |
norman | \(\left (-\frac {15}{2}+8 x -\frac {1}{2} x^{2}\right ) {\mathrm e}^{-x}\) | \(16\) |
risch | \(\frac {\left (-x^{2}+16 x -15\right ) {\mathrm e}^{-x}}{2}\) | \(17\) |
default | \(-\frac {x^{2} {\mathrm e}^{-x}}{2}+8 x \,{\mathrm e}^{-x}-\frac {15 \,{\mathrm e}^{-x}}{2}\) | \(24\) |
meijerg | \(\frac {15}{2}-\frac {31 \,{\mathrm e}^{-x}}{2}-\frac {\left (3 x^{2}+6 x +6\right ) {\mathrm e}^{-x}}{6}+\frac {9 \left (2 x +2\right ) {\mathrm e}^{-x}}{2}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 30, normalized size = 1.15 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + 9 \, {\left (x + 1\right )} e^{\left (-x\right )} - \frac {31}{2} \, e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 12, normalized size = 0.46 \begin {gather*} -\frac {{\mathrm {e}}^{-x}\,\left (x-1\right )\,\left (x-15\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 12, normalized size = 0.46 \begin {gather*} \frac {\left (- x^{2} + 16 x - 15\right ) e^{- x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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