Integrand size = 40, antiderivative size = 29 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=-8-e^{3 (-5+x)}+\frac {1}{3} e^{\frac {-e+x+x^2}{x}} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 14, 2225, 6838} \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=\frac {1}{3} e^{x-\frac {e}{x}+1}-e^{3 x-15} \]
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Rule 12
Rule 14
Rule 2225
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{x^2} \, dx \\ & = \frac {1}{3} \int \left (-9 e^{-15+3 x}+\frac {e^{1-\frac {e}{x}+x} \left (e+x^2\right )}{x^2}\right ) \, dx \\ & = \frac {1}{3} \int \frac {e^{1-\frac {e}{x}+x} \left (e+x^2\right )}{x^2} \, dx-3 \int e^{-15+3 x} \, dx \\ & = \frac {1}{3} e^{1-\frac {e}{x}+x}-e^{-15+3 x} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=\frac {1}{3} e^{1-\frac {e}{x}+x}-e^{-15+3 x} \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
parts | \(-{\mathrm e}^{3 x -15}+\frac {{\mathrm e}^{\frac {-{\mathrm e}+x^{2}+x}{x}}}{3}\) | \(26\) |
risch | \(-{\mathrm e}^{3 x -15}+\frac {{\mathrm e}^{-\frac {-x^{2}+{\mathrm e}-x}{x}}}{3}\) | \(29\) |
parallelrisch | \(-{\mathrm e}^{3 x -15}+\frac {{\mathrm e}^{-\frac {-x^{2}+{\mathrm e}-x}{x}}}{3}\) | \(29\) |
norman | \(\frac {\frac {x \,{\mathrm e}^{\frac {-{\mathrm e}+x^{2}+x}{x}}}{3}-x \,{\mathrm e}^{3 x -15}}{x}\) | \(32\) |
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=-e^{\left (3 \, x - 15\right )} + \frac {1}{3} \, e^{\left (\frac {x^{2} + x - e}{x}\right )} \]
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Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=\frac {e^{\frac {x^{2} + x - e}{x}}}{3} - e^{3 x - 15} \]
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=-\frac {1}{3} \, {\left (3 \, e^{\left (3 \, x\right )} - e^{\left (x - \frac {e}{x} + 16\right )}\right )} e^{\left (-15\right )} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=-\frac {1}{3} \, {\left (3 \, e^{\left (3 \, x\right )} - e^{\left (\frac {x^{2} + x - e}{x} + 15\right )}\right )} e^{\left (-15\right )} \]
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Time = 8.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {-9 e^{-15+3 x} x^2+e^{\frac {-e+x+x^2}{x}} \left (e+x^2\right )}{3 x^2} \, dx=\frac {{\mathrm {e}}^{-\frac {\mathrm {e}}{x}}\,\mathrm {e}\,{\mathrm {e}}^x}{3}-{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-15} \]
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