Integrand size = 39, antiderivative size = 22 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=-6+e^x+5 \left (2+15 e^{-2 x}+e^x\right ) x^2 \]
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Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 17, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6820, 2227, 2207, 2225} \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=75 e^{-2 x} x^2+5 e^x x^2+10 x^2+e^x \]
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Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (20 x-150 e^{-2 x} (-1+x) x+e^x \left (1+10 x+5 x^2\right )\right ) \, dx \\ & = 10 x^2-150 \int e^{-2 x} (-1+x) x \, dx+\int e^x \left (1+10 x+5 x^2\right ) \, dx \\ & = 10 x^2-150 \int \left (-e^{-2 x} x+e^{-2 x} x^2\right ) \, dx+\int \left (e^x+10 e^x x+5 e^x x^2\right ) \, dx \\ & = 10 x^2+5 \int e^x x^2 \, dx+10 \int e^x x \, dx+150 \int e^{-2 x} x \, dx-150 \int e^{-2 x} x^2 \, dx+\int e^x \, dx \\ & = e^x-75 e^{-2 x} x+10 e^x x+10 x^2+75 e^{-2 x} x^2+5 e^x x^2-10 \int e^x \, dx-10 \int e^x x \, dx+75 \int e^{-2 x} \, dx-150 \int e^{-2 x} x \, dx \\ & = -\frac {75}{2} e^{-2 x}-9 e^x+10 x^2+75 e^{-2 x} x^2+5 e^x x^2+10 \int e^x \, dx-75 \int e^{-2 x} \, dx \\ & = e^x+10 x^2+75 e^{-2 x} x^2+5 e^x x^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=10 x^2+75 e^{-2 x} x^2+e^x \left (1+5 x^2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
default | \(5 \,{\mathrm e}^{x} x^{2}+75 \,{\mathrm e}^{-2 x} x^{2}+10 x^{2}+{\mathrm e}^{x}\) | \(25\) |
parts | \(5 \,{\mathrm e}^{x} x^{2}+75 \,{\mathrm e}^{-2 x} x^{2}+10 x^{2}+{\mathrm e}^{x}\) | \(25\) |
risch | \(10 x^{2}+\left (5 x^{2}+1\right ) {\mathrm e}^{x}+75 \,{\mathrm e}^{-2 x} x^{2}\) | \(26\) |
norman | \(\left ({\mathrm e}^{3 x}+75 x^{2}+5 x^{2} {\mathrm e}^{3 x}+10 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-2 x}\) | \(34\) |
parallelrisch | \(\left ({\mathrm e}^{3 x}+75 x^{2}+5 x^{2} {\mathrm e}^{3 x}+10 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-2 x}\) | \(34\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx={\left (10 \, x^{2} e^{\left (2 \, x\right )} + 75 \, x^{2} + {\left (5 \, x^{2} + 1\right )} e^{\left (3 \, x\right )}\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=10 x^{2} + 75 x^{2} e^{- 2 x} + \left (5 x^{2} + 1\right ) e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=10 \, x^{2} + \frac {75}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {75}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 5 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 10 \, {\left (x - 1\right )} e^{x} + e^{x} \]
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=75 \, x^{2} e^{\left (-2 \, x\right )} + 10 \, x^{2} + {\left (5 \, x^{2} + 1\right )} e^{x} \]
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Time = 13.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int e^{-2 x} \left (150 x+20 e^{2 x} x-150 x^2+e^{3 x} \left (1+10 x+5 x^2\right )\right ) \, dx=75\,x^2\,{\mathrm {e}}^{-2\,x}+{\mathrm {e}}^x\,\left (5\,x^2+1\right )+10\,x^2 \]
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