Optimal. Leaf size=22 \[ e^{-x} \left (-1+x^2+(20+x) \left (1+e^5 x\right )\right ) \]
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Rubi [B] time = 0.11, antiderivative size = 45, normalized size of antiderivative = 2.05, number of steps used = 16, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2196, 2194, 2176} \begin {gather*} e^{5-x} x^2+e^{-x} x^2+20 e^{5-x} x+e^{-x} x+19 e^{-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-18 e^{-x}+e^{-x} x-e^{-x} x^2-e^{5-x} \left (-20+18 x+x^2\right )\right ) \, dx\\ &=-\left (18 \int e^{-x} \, dx\right )+\int e^{-x} x \, dx-\int e^{-x} x^2 \, dx-\int e^{5-x} \left (-20+18 x+x^2\right ) \, dx\\ &=18 e^{-x}-e^{-x} x+e^{-x} x^2-2 \int e^{-x} x \, dx+\int e^{-x} \, dx-\int \left (-20 e^{5-x}+18 e^{5-x} x+e^{5-x} x^2\right ) \, dx\\ &=17 e^{-x}+e^{-x} x+e^{-x} x^2-2 \int e^{-x} \, dx-18 \int e^{5-x} x \, dx+20 \int e^{5-x} \, dx-\int e^{5-x} x^2 \, dx\\ &=-20 e^{5-x}+19 e^{-x}+18 e^{5-x} x+e^{-x} x+e^{5-x} x^2+e^{-x} x^2-2 \int e^{5-x} x \, dx-18 \int e^{5-x} \, dx\\ &=-2 e^{5-x}+19 e^{-x}+20 e^{5-x} x+e^{-x} x+e^{5-x} x^2+e^{-x} x^2-2 \int e^{5-x} \, dx\\ &=19 e^{-x}+20 e^{5-x} x+e^{-x} x+e^{5-x} x^2+e^{-x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 26, normalized size = 1.18 \begin {gather*} e^{-x} \left (19+\left (1+20 e^5\right ) x+\left (1+e^5\right ) x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 21, normalized size = 0.95 \begin {gather*} {\left (x^{2} + {\left (x^{2} + 20 \, x\right )} e^{5} + x + 19\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 26, normalized size = 1.18 \begin {gather*} {\left (x^{2} + x + 19\right )} e^{\left (-x\right )} + {\left (x^{2} + 20 \, x\right )} e^{\left (-x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 1.05
method | result | size |
gosper | \(\left (x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}+x^{2}+x +19\right ) {\mathrm e}^{-x}\) | \(23\) |
risch | \(\left (x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}+x^{2}+x +19\right ) {\mathrm e}^{-x}\) | \(23\) |
norman | \(\left (19+\left ({\mathrm e}^{5}+1\right ) x^{2}+\left (20 \,{\mathrm e}^{5}+1\right ) x \right ) {\mathrm e}^{-x}\) | \(24\) |
meijerg | \(\left (-{\mathrm e}^{5}-1\right ) \left (2-\frac {\left (3 x^{2}+6 x +6\right ) {\mathrm e}^{-x}}{3}\right )+\left (-18 \,{\mathrm e}^{5}+1\right ) \left (1-\frac {\left (2 x +2\right ) {\mathrm e}^{-x}}{2}\right )+20 \,{\mathrm e}^{5} \left (1-{\mathrm e}^{-x}\right )-18+18 \,{\mathrm e}^{-x}\) | \(66\) |
default | \(x \,{\mathrm e}^{-x}+19 \,{\mathrm e}^{-x}+x^{2} {\mathrm e}^{-x}-20 \,{\mathrm e}^{5} {\mathrm e}^{-x}-18 \,{\mathrm e}^{5} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )-{\mathrm e}^{5} \left (-x^{2} {\mathrm e}^{-x}-2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}\right )\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 71, normalized size = 3.23 \begin {gather*} {\left (x^{2} e^{5} + 2 \, x e^{5} + 2 \, e^{5}\right )} e^{\left (-x\right )} + {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + 18 \, {\left (x e^{5} + e^{5}\right )} e^{\left (-x\right )} - {\left (x + 1\right )} e^{\left (-x\right )} + 18 \, e^{\left (-x\right )} - 20 \, e^{\left (-x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 22, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{-x}\,\left (x+20\,x\,{\mathrm {e}}^5+x^2\,{\mathrm {e}}^5+x^2+19\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 22, normalized size = 1.00 \begin {gather*} \left (x^{2} + x^{2} e^{5} + x + 20 x e^{5} + 19\right ) e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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