Integrand size = 25, antiderivative size = 13 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=-5+e^{\left (-1+\frac {9 x}{25}\right )^2} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 2259, 2240} \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\frac {1}{625} (25-9 x)^2} \]
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Rule 12
Rule 2240
Rule 2259
Rubi steps \begin{align*} \text {integral}& = \frac {1}{625} \int e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx \\ & = \frac {1}{625} \int e^{\frac {1}{625} (-25+9 x)^2} (-450+162 x) \, dx \\ & = e^{\frac {1}{625} (25-9 x)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\frac {1}{625} (25-9 x)^2} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85
method | result | size |
risch | \({\mathrm e}^{\frac {\left (9 x -25\right )^{2}}{625}}\) | \(11\) |
gosper | \({\mathrm e}^{\frac {81}{625} x^{2}-\frac {18}{25} x +1}\) | \(12\) |
default | \({\mathrm e}^{\frac {81}{625} x^{2}-\frac {18}{25} x +1}\) | \(12\) |
norman | \({\mathrm e}^{\frac {81}{625} x^{2}-\frac {18}{25} x +1}\) | \(12\) |
parallelrisch | \({\mathrm e}^{\frac {81}{625} x^{2}-\frac {18}{25} x +1}\) | \(12\) |
parts | \(-\frac {9 i \sqrt {\pi }\, \operatorname {erf}\left (\frac {9}{25} i x -i\right ) x}{25}+i \sqrt {\pi }\, \operatorname {erf}\left (\frac {9}{25} i x -i\right )+\sqrt {\pi }\, \left (\operatorname {erf}\left (\frac {9}{25} i x -i\right ) \left (\frac {9}{25} i x -i\right )+\frac {{\mathrm e}^{-\left (\frac {9}{25} i x -i\right )^{2}}}{\sqrt {\pi }}\right )\) | \(68\) |
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Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\left (\frac {81}{625} \, x^{2} - \frac {18}{25} \, x + 1\right )} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\frac {81 x^{2}}{625} - \frac {18 x}{25} + 1} \]
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Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\left (\frac {81}{625} \, x^{2} - \frac {18}{25} \, x + 1\right )} \]
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx=e^{\left (\frac {81}{625} \, x^{2} - \frac {18}{25} \, x + 1\right )} \]
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Time = 10.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{625} e^{\frac {1}{625} \left (625-450 x+81 x^2\right )} (-450+162 x) \, dx={\mathrm {e}}^{-\frac {18\,x}{25}}\,\mathrm {e}\,{\mathrm {e}}^{\frac {81\,x^2}{625}} \]
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