Integrand size = 33, antiderivative size = 27 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=5-e^3+\frac {e^2}{4 x^2}+2 x+\left (5+e^x\right ) x \]
[Out]
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {12, 14, 2207, 2225} \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=\frac {e^2}{4 x^2}+7 x-e^x+e^x (x+1) \]
[In]
[Out]
Rule 12
Rule 14
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{x^3} \, dx \\ & = \frac {1}{2} \int \left (2 e^x (1+x)+\frac {-e^2+14 x^3}{x^3}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-e^2+14 x^3}{x^3} \, dx+\int e^x (1+x) \, dx \\ & = e^x (1+x)+\frac {1}{2} \int \left (14-\frac {e^2}{x^3}\right ) \, dx-\int e^x \, dx \\ & = -e^x+\frac {e^2}{4 x^2}+7 x+e^x (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=\frac {e^2}{4 x^2}+7 x+e^x x \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59
method | result | size |
default | \(7 x +\frac {{\mathrm e}^{2}}{4 x^{2}}+{\mathrm e}^{x} x\) | \(16\) |
risch | \(7 x +\frac {{\mathrm e}^{2}}{4 x^{2}}+{\mathrm e}^{x} x\) | \(16\) |
parts | \(7 x +\frac {{\mathrm e}^{2}}{4 x^{2}}+{\mathrm e}^{x} x\) | \(16\) |
norman | \(\frac {{\mathrm e}^{x} x^{3}+7 x^{3}+\frac {{\mathrm e}^{2}}{4}}{x^{2}}\) | \(21\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{x} x^{3}+28 x^{3}+{\mathrm e}^{2}}{4 x^{2}}\) | \(21\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=\frac {4 \, x^{3} e^{x} + 28 \, x^{3} + e^{2}}{4 \, x^{2}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=x e^{x} + 7 x + \frac {e^{2}}{4 x^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx={\left (x - 1\right )} e^{x} + 7 \, x + \frac {e^{2}}{4 \, x^{2}} + e^{x} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=\frac {4 \, x^{3} e^{x} + 28 \, x^{3} + e^{2}}{4 \, x^{2}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {-e^2+14 x^3+e^x \left (2 x^3+2 x^4\right )}{2 x^3} \, dx=x\,\left ({\mathrm {e}}^x+7\right )+\frac {{\mathrm {e}}^2}{4\,x^2} \]
[In]
[Out]