Integrand size = 28, antiderivative size = 27 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {1}{5} \left (-5-e^3+x+6 \left (-4+e^5-e^x\right ) x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6, 12, 1607, 2227, 2207, 2225} \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=-\frac {6}{5} e^x x^2-\frac {6}{5} \left (4-e^5\right ) x^2+\frac {x}{5} \]
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Rule 6
Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{5} \left (1+\left (-48+12 e^5\right ) x+e^x \left (-12 x-6 x^2\right )\right ) \, dx \\ & = \frac {1}{5} \int \left (1+\left (-48+12 e^5\right ) x+e^x \left (-12 x-6 x^2\right )\right ) \, dx \\ & = \frac {x}{5}-\frac {6}{5} \left (4-e^5\right ) x^2+\frac {1}{5} \int e^x \left (-12 x-6 x^2\right ) \, dx \\ & = \frac {x}{5}-\frac {6}{5} \left (4-e^5\right ) x^2+\frac {1}{5} \int e^x (-12-6 x) x \, dx \\ & = \frac {x}{5}-\frac {6}{5} \left (4-e^5\right ) x^2+\frac {1}{5} \int \left (-12 e^x x-6 e^x x^2\right ) \, dx \\ & = \frac {x}{5}-\frac {6}{5} \left (4-e^5\right ) x^2-\frac {6}{5} \int e^x x^2 \, dx-\frac {12}{5} \int e^x x \, dx \\ & = \frac {x}{5}-\frac {12 e^x x}{5}-\frac {6 e^x x^2}{5}-\frac {6}{5} \left (4-e^5\right ) x^2+\frac {12 \int e^x \, dx}{5}+\frac {12}{5} \int e^x x \, dx \\ & = \frac {12 e^x}{5}+\frac {x}{5}-\frac {6 e^x x^2}{5}-\frac {6}{5} \left (4-e^5\right ) x^2-\frac {12 \int e^x \, dx}{5} \\ & = \frac {x}{5}-\frac {6 e^x x^2}{5}-\frac {6}{5} \left (4-e^5\right ) x^2 \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {1}{5} \left (x-6 \left (4-e^5+e^x\right ) x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\left (\frac {6 \,{\mathrm e}^{5}}{5}-\frac {24}{5}\right ) x^{2}+\frac {x}{5}-\frac {6 \,{\mathrm e}^{x} x^{2}}{5}\) | \(22\) |
default | \(\frac {x}{5}-\frac {6 \,{\mathrm e}^{x} x^{2}}{5}-\frac {24 x^{2}}{5}+\frac {6 x^{2} {\mathrm e}^{5}}{5}\) | \(24\) |
risch | \(\frac {x}{5}-\frac {6 \,{\mathrm e}^{x} x^{2}}{5}-\frac {24 x^{2}}{5}+\frac {6 x^{2} {\mathrm e}^{5}}{5}\) | \(24\) |
parallelrisch | \(\frac {x}{5}-\frac {6 \,{\mathrm e}^{x} x^{2}}{5}-\frac {24 x^{2}}{5}+\frac {6 x^{2} {\mathrm e}^{5}}{5}\) | \(24\) |
parts | \(\frac {x}{5}-\frac {6 \,{\mathrm e}^{x} x^{2}}{5}-\frac {24 x^{2}}{5}+\frac {6 x^{2} {\mathrm e}^{5}}{5}\) | \(24\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {6}{5} \, x^{2} e^{5} - \frac {6}{5} \, x^{2} e^{x} - \frac {24}{5} \, x^{2} + \frac {1}{5} \, x \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=- \frac {6 x^{2} e^{x}}{5} + x^{2} \left (- \frac {24}{5} + \frac {6 e^{5}}{5}\right ) + \frac {x}{5} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {6}{5} \, x^{2} e^{5} - \frac {6}{5} \, x^{2} e^{x} - \frac {24}{5} \, x^{2} + \frac {1}{5} \, x \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {6}{5} \, x^{2} e^{5} - \frac {6}{5} \, x^{2} e^{x} - \frac {24}{5} \, x^{2} + \frac {1}{5} \, x \]
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Time = 10.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1}{5} \left (1-48 x+12 e^5 x+e^x \left (-12 x-6 x^2\right )\right ) \, dx=\frac {x}{5}-\frac {6\,x^2\,{\mathrm {e}}^x}{5}+x^2\,\left (\frac {6\,{\mathrm {e}}^5}{5}-\frac {24}{5}\right ) \]
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