Optimal. Leaf size=26 \[ 9 \left (2+5 e^{-\frac {x}{2}+x \left (-\frac {4}{x}+x\right )}+x\right )^2 \]
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Rubi [A] time = 0.63, antiderivative size = 52, normalized size of antiderivative = 2.00, number of steps used = 19, number of rules used = 11, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1584, 2236, 1593, 2274, 12, 6742, 2244, 2240, 2234, 2204, 2241} \begin {gather*} 9 x^2+90 e^{x^2-\frac {x}{2}-4} x+180 e^{x^2-\frac {x}{2}-4}+225 e^{2 x^2-x-8}+36 x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1584
Rule 1593
Rule 2204
Rule 2234
Rule 2236
Rule 2240
Rule 2241
Rule 2244
Rule 2274
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=36 x+9 x^2+25 \int \frac {e^{-8-x+2 x^2} \left (-9 x^2+36 x^3\right )}{x^2} \, dx+\int e^{\frac {1}{2} \left (-8-x+2 x^2+2 \log \left (\frac {5}{x}\right )\right )} \left (63 x^2+36 x^3\right ) \, dx\\ &=36 x+9 x^2+25 \int e^{-8-x+2 x^2} (-9+36 x) \, dx+\int e^{\frac {1}{2} \left (-8-x+2 x^2+2 \log \left (\frac {5}{x}\right )\right )} x^2 (63+36 x) \, dx\\ &=225 e^{-8-x+2 x^2}+36 x+9 x^2+\int 5 e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x (63+36 x) \, dx\\ &=225 e^{-8-x+2 x^2}+36 x+9 x^2+5 \int e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x (63+36 x) \, dx\\ &=225 e^{-8-x+2 x^2}+36 x+9 x^2+5 \int \left (63 e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x+36 e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x^2\right ) \, dx\\ &=225 e^{-8-x+2 x^2}+36 x+9 x^2+180 \int e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x^2 \, dx+315 \int e^{\frac {1}{2} \left (-8-x+2 x^2\right )} x \, dx\\ &=225 e^{-8-x+2 x^2}+36 x+9 x^2+180 \int e^{-4-\frac {x}{2}+x^2} x^2 \, dx+315 \int e^{-4-\frac {x}{2}+x^2} x \, dx\\ &=\frac {315}{2} e^{-4-\frac {x}{2}+x^2}+225 e^{-8-x+2 x^2}+36 x+90 e^{-4-\frac {x}{2}+x^2} x+9 x^2+45 \int e^{-4-\frac {x}{2}+x^2} x \, dx+\frac {315}{4} \int e^{-4-\frac {x}{2}+x^2} \, dx-90 \int e^{-4-\frac {x}{2}+x^2} \, dx\\ &=180 e^{-4-\frac {x}{2}+x^2}+225 e^{-8-x+2 x^2}+36 x+90 e^{-4-\frac {x}{2}+x^2} x+9 x^2+\frac {45}{4} \int e^{-4-\frac {x}{2}+x^2} \, dx+\frac {315 \int e^{\frac {1}{4} \left (-\frac {1}{2}+2 x\right )^2} \, dx}{4 e^{65/16}}-\frac {90 \int e^{\frac {1}{4} \left (-\frac {1}{2}+2 x\right )^2} \, dx}{e^{65/16}}\\ &=180 e^{-4-\frac {x}{2}+x^2}+225 e^{-8-x+2 x^2}+36 x+90 e^{-4-\frac {x}{2}+x^2} x+9 x^2-\frac {45 \sqrt {\pi } \text {erfi}\left (\frac {1}{4} (-1+4 x)\right )}{8 e^{65/16}}+\frac {45 \int e^{\frac {1}{4} \left (-\frac {1}{2}+2 x\right )^2} \, dx}{4 e^{65/16}}\\ &=180 e^{-4-\frac {x}{2}+x^2}+225 e^{-8-x+2 x^2}+36 x+90 e^{-4-\frac {x}{2}+x^2} x+9 x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 48, normalized size = 1.85 \begin {gather*} 9 \left (25 e^{-8-x+2 x^2}+4 x+x^2+5 e^{-\frac {x}{2}+x^2} \left (\frac {4}{e^4}+\frac {2 x}{e^4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 57, normalized size = 2.19 \begin {gather*} 9 \, x^{2} e^{\left (2 \, x^{2} - x + 2 \, \log \left (\frac {5}{x}\right ) - 8\right )} + 9 \, x^{2} + 18 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x^{2} - \frac {1}{2} \, x + \log \left (\frac {5}{x}\right ) - 4\right )} + 36 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 47, normalized size = 1.81 \begin {gather*} 9 \, x^{2} + 90 \, {\left (x e^{\left (x^{2} - \frac {1}{2} \, x\right )} + 2 \, e^{\left (x^{2} - \frac {1}{2} \, x\right )}\right )} e^{\left (-4\right )} + 36 \, x + 225 \, e^{\left (2 \, x^{2} - x - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 39, normalized size = 1.50
method | result | size |
risch | \(225 \,{\mathrm e}^{2 x^{2}-x -8}+5 \left (36+18 x \right ) {\mathrm e}^{-4+x^{2}-\frac {1}{2} x}+9 x^{2}+36 x\) | \(39\) |
norman | \(36 x +9 x^{2}+36 x \,{\mathrm e}^{\ln \left (\frac {5}{x}\right )+x^{2}-\frac {x}{2}-4}+18 x^{2} {\mathrm e}^{\ln \left (\frac {5}{x}\right )+x^{2}-\frac {x}{2}-4}+225 \,{\mathrm e}^{2 x^{2}-x -8}\) | \(70\) |
default | \(36 x +9 \,{\mathrm e}^{2 x^{2}-x +2 \ln \left (\frac {5}{x}\right )+2 \ln \relax (x )-8}+36 \,{\mathrm e}^{-4+\ln \left (\frac {5}{x}\right )+\ln \relax (x )+x^{2}-\frac {x}{2}}+18 x \,{\mathrm e}^{-4+\ln \left (\frac {5}{x}\right )+\ln \relax (x )+x^{2}-\frac {x}{2}}+9 x^{2}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 36, normalized size = 1.38 \begin {gather*} 9 \, x^{2} + 90 \, {\left (x + 2\right )} e^{\left (x^{2} - \frac {1}{2} \, x - 4\right )} + 36 \, x + 225 \, e^{\left (2 \, x^{2} - x - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 45, normalized size = 1.73 \begin {gather*} 36\,x+180\,{\mathrm {e}}^{x^2-\frac {x}{2}-4}+225\,{\mathrm {e}}^{2\,x^2-x-8}+90\,x\,{\mathrm {e}}^{x^2-\frac {x}{2}-4}+9\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 34, normalized size = 1.31 \begin {gather*} 9 x^{2} + 36 x + \left (90 x + 180\right ) e^{x^{2} - \frac {x}{2} - 4} + 225 e^{2 x^{2} - x - 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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