Integrand size = 25, antiderivative size = 27 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=\frac {1}{2} \left (e x+5 \left (6 x-3 \left (1+e^{-x} x^2\right )\right )\right ) \]
[Out]
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 6820, 2227, 2207, 2225} \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=\frac {1}{2} (30+e) x-\frac {15}{2} e^{-x} x^2 \]
[In]
[Out]
Rule 12
Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx \\ & = \frac {1}{2} \int \left (30+e+15 e^{-x} (-2+x) x\right ) \, dx \\ & = \frac {1}{2} (30+e) x+\frac {15}{2} \int e^{-x} (-2+x) x \, dx \\ & = \frac {1}{2} (30+e) x+\frac {15}{2} \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx \\ & = \frac {1}{2} (30+e) x+\frac {15}{2} \int e^{-x} x^2 \, dx-15 \int e^{-x} x \, dx \\ & = 15 e^{-x} x+\frac {1}{2} (30+e) x-\frac {15}{2} e^{-x} x^2-15 \int e^{-x} \, dx+15 \int e^{-x} x \, dx \\ & = 15 e^{-x}+\frac {1}{2} (30+e) x-\frac {15}{2} e^{-x} x^2+15 \int e^{-x} \, dx \\ & = \frac {1}{2} (30+e) x-\frac {15}{2} e^{-x} x^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=\frac {1}{2} \left (30 x+e x-15 e^{-x} x^2\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {x \,{\mathrm e}}{2}+15 x -\frac {15 x^{2} {\mathrm e}^{-x}}{2}\) | \(19\) |
risch | \(\frac {x \,{\mathrm e}}{2}+15 x -\frac {15 x^{2} {\mathrm e}^{-x}}{2}\) | \(19\) |
parts | \(\frac {x \,{\mathrm e}}{2}+15 x -\frac {15 x^{2} {\mathrm e}^{-x}}{2}\) | \(19\) |
norman | \(\left (\left (\frac {{\mathrm e}}{2}+15\right ) x \,{\mathrm e}^{x}-\frac {15 x^{2}}{2}\right ) {\mathrm e}^{-x}\) | \(22\) |
parallelrisch | \(\frac {\left (x \,{\mathrm e} \,{\mathrm e}^{x}-15 x^{2}+30 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{-x}}{2}\) | \(24\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=-\frac {1}{2} \, {\left (15 \, x^{2} - {\left (x e + 30 \, x\right )} e^{x}\right )} e^{\left (-x\right )} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=- \frac {15 x^{2} e^{- x}}{2} + x \left (\frac {e}{2} + 15\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=\frac {1}{2} \, x e - \frac {15}{2} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + 15 \, {\left (x + 1\right )} e^{\left (-x\right )} + 15 \, x \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=-\frac {15}{2} \, x^{2} e^{\left (-x\right )} + \frac {1}{2} \, x e + 15 \, x \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {1}{2} e^{-x} \left (e^x (30+e)-30 x+15 x^2\right ) \, dx=\frac {x\,\left (\mathrm {e}-15\,x\,{\mathrm {e}}^{-x}+30\right )}{2} \]
[In]
[Out]