Integrand size = 134, antiderivative size = 30 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {2 e^6 (3+x)^2}{e^x+\frac {1}{15} (x-\log (5-x))} \]
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\[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {30 e^6 (3+x) \left (-18+7 x-x^2+15 e^x \left (-5-4 x+x^2\right )+2 (-5+x) \log (5-x)\right )}{(5-x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = \left (30 e^6\right ) \int \frac {(3+x) \left (-18+7 x-x^2+15 e^x \left (-5-4 x+x^2\right )+2 (-5+x) \log (5-x)\right )}{(5-x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = \left (30 e^6\right ) \int \left (-\frac {3+4 x+x^2}{15 e^x+x-\log (5-x)}+\frac {(3+x)^2 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx \\ & = -\left (\left (30 e^6\right ) \int \frac {3+4 x+x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \frac {(3+x)^2 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = -\left (\left (30 e^6\right ) \int \left (\frac {3}{15 e^x+x-\log (5-x)}+\frac {4 x}{15 e^x+x-\log (5-x)}+\frac {x^2}{15 e^x+x-\log (5-x)}\right ) \, dx\right )+\left (30 e^6\right ) \int \left (\frac {11 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {64 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx \\ & = -\left (\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \frac {x \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (330 e^6\right ) \int \frac {6-6 x+x^2+5 \log (5-x)-x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {6-6 x+x^2+5 \log (5-x)-x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = -\left (\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \left (\frac {6 x}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x^2}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (330 e^6\right ) \int \left (\frac {6}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (1920 e^6\right ) \int \left (\frac {6}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^2}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}-\frac {x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx \\ & = \left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {x^2}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1920 e^6\right ) \int \frac {x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {\log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \frac {x}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = \left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \left (\frac {5}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {25}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx-\left (1920 e^6\right ) \int \left (\frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {\log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \left (\frac {1}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ & = \left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1920 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (48000 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (57600 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=-\frac {30 e^6 (3+x)^2}{-15 e^x-x+\log (5-x)} \]
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {30 \left (x^{2}+6 x +9\right ) {\mathrm e}^{6}}{15 \,{\mathrm e}^{x}-\ln \left (5-x \right )+x}\) | \(29\) |
parallelrisch | \(\frac {30 x^{2} {\mathrm e}^{6}+180 x \,{\mathrm e}^{6}+270 \,{\mathrm e}^{6}}{15 \,{\mathrm e}^{x}-\ln \left (5-x \right )+x}\) | \(35\) |
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left (x^{2} + 6 \, x + 9\right )} e^{12}}{x e^{6} - e^{6} \log \left (-x + 5\right ) + 15 \, e^{\left (x + 6\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {2 x^{2} e^{6} + 12 x e^{6} + 18 e^{6}}{\frac {x}{15} + e^{x} - \frac {\log {\left (5 - x \right )}}{15}} \]
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left (x^{2} e^{6} + 6 \, x e^{6} + 9 \, e^{6}\right )}}{x + 15 \, e^{x} - \log \left (-x + 5\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 628, normalized size of antiderivative = 20.93 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left ({\left (x - 5\right )}^{5} e^{\left (-x + 11\right )} - 2 \, {\left (x - 5\right )}^{4} e^{\left (-x + 11\right )} \log \left (-x + 5\right ) + {\left (x - 5\right )}^{3} e^{\left (-x + 11\right )} \log \left (-x + 5\right )^{2} + 15 \, {\left (x - 5\right )}^{4} e^{11} + 25 \, {\left (x - 5\right )}^{4} e^{\left (-x + 11\right )} - 15 \, {\left (x - 5\right )}^{3} e^{11} \log \left (-x + 5\right ) - 41 \, {\left (x - 5\right )}^{3} e^{\left (-x + 11\right )} \log \left (-x + 5\right ) + 16 \, {\left (x - 5\right )}^{2} e^{\left (-x + 11\right )} \log \left (-x + 5\right )^{2} + 300 \, {\left (x - 5\right )}^{3} e^{11} + 229 \, {\left (x - 5\right )}^{3} e^{\left (-x + 11\right )} - 240 \, {\left (x - 5\right )}^{2} e^{11} \log \left (-x + 5\right ) - 273 \, {\left (x - 5\right )}^{2} e^{\left (-x + 11\right )} \log \left (-x + 5\right ) + 64 \, {\left (x - 5\right )} e^{\left (-x + 11\right )} \log \left (-x + 5\right )^{2} + 1935 \, {\left (x - 5\right )}^{2} e^{11} + 917 \, {\left (x - 5\right )}^{2} e^{\left (-x + 11\right )} - 960 \, {\left (x - 5\right )} e^{11} \log \left (-x + 5\right ) - 592 \, {\left (x - 5\right )} e^{\left (-x + 11\right )} \log \left (-x + 5\right ) + 4080 \, {\left (x - 5\right )} e^{11} + 1424 \, {\left (x - 5\right )} e^{\left (-x + 11\right )} - 64 \, e^{\left (-x + 11\right )} \log \left (-x + 5\right ) + 960 \, e^{11} + 320 \, e^{\left (-x + 11\right )}\right )}}{{\left (x - 5\right )}^{4} e^{\left (-x + 5\right )} - 3 \, {\left (x - 5\right )}^{3} e^{\left (-x + 5\right )} \log \left (-x + 5\right ) + 3 \, {\left (x - 5\right )}^{2} e^{\left (-x + 5\right )} \log \left (-x + 5\right )^{2} - {\left (x - 5\right )} e^{\left (-x + 5\right )} \log \left (-x + 5\right )^{3} + 30 \, {\left (x - 5\right )}^{3} e^{5} + 14 \, {\left (x - 5\right )}^{3} e^{\left (-x + 5\right )} - 60 \, {\left (x - 5\right )}^{2} e^{5} \log \left (-x + 5\right ) - 28 \, {\left (x - 5\right )}^{2} e^{\left (-x + 5\right )} \log \left (-x + 5\right ) + 30 \, {\left (x - 5\right )} e^{5} \log \left (-x + 5\right )^{2} + 14 \, {\left (x - 5\right )} e^{\left (-x + 5\right )} \log \left (-x + 5\right )^{2} + 270 \, {\left (x - 5\right )}^{2} e^{5} + 225 \, {\left (x - 5\right )}^{2} e^{\left (x + 5\right )} + 66 \, {\left (x - 5\right )}^{2} e^{\left (-x + 5\right )} - 270 \, {\left (x - 5\right )} e^{5} \log \left (-x + 5\right ) - 225 \, {\left (x - 5\right )} e^{\left (x + 5\right )} \log \left (-x + 5\right ) - 67 \, {\left (x - 5\right )} e^{\left (-x + 5\right )} \log \left (-x + 5\right ) + e^{\left (-x + 5\right )} \log \left (-x + 5\right )^{2} + 630 \, {\left (x - 5\right )} e^{5} + 900 \, {\left (x - 5\right )} e^{\left (x + 5\right )} + 110 \, {\left (x - 5\right )} e^{\left (-x + 5\right )} - 30 \, e^{5} \log \left (-x + 5\right ) - 10 \, e^{\left (-x + 5\right )} \log \left (-x + 5\right ) + 150 \, e^{5} + 225 \, e^{\left (x + 5\right )} + 25 \, e^{\left (-x + 5\right )}} \]
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Timed out. \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\int \frac {{\mathrm {e}}^6\,{\mathrm {e}}^x\,\left (-450\,x^3+450\,x^2+7650\,x+6750\right )-{\mathrm {e}}^6\,\left (-30\,x^3+120\,x^2+90\,x-1620\right )+{\mathrm {e}}^6\,\ln \left (5-x\right )\,\left (-60\,x^2+120\,x+900\right )}{{\ln \left (5-x\right )}^2\,\left (x-5\right )-{\mathrm {e}}^x\,\left (150\,x-30\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (225\,x-1125\right )-5\,x^2+x^3-\ln \left (5-x\right )\,\left ({\mathrm {e}}^x\,\left (30\,x-150\right )-10\,x+2\,x^2\right )} \,d x \]
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