Integrand size = 33, antiderivative size = 20 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=-5+\left (-2-\frac {1}{3} e^{2+\frac {x}{2}} x\right )^2 \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(20)=40\).
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 2207, 2225, 1607, 2227} \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {1}{9} e^{x+4} x^2-\frac {8}{3} e^{\frac {x}{2}+2}+\frac {4}{3} e^{\frac {x}{2}+2} (x+2) \]
[In]
[Out]
Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx \\ & = \frac {1}{9} \int e^{2+\frac {x}{2}} (12+6 x) \, dx+\frac {1}{9} \int e^{4+x} \left (2 x+x^2\right ) \, dx \\ & = \frac {4}{3} e^{2+\frac {x}{2}} (2+x)+\frac {1}{9} \int e^{4+x} x (2+x) \, dx-\frac {4}{3} \int e^{2+\frac {x}{2}} \, dx \\ & = -\frac {8}{3} e^{2+\frac {x}{2}}+\frac {4}{3} e^{2+\frac {x}{2}} (2+x)+\frac {1}{9} \int \left (2 e^{4+x} x+e^{4+x} x^2\right ) \, dx \\ & = -\frac {8}{3} e^{2+\frac {x}{2}}+\frac {4}{3} e^{2+\frac {x}{2}} (2+x)+\frac {1}{9} \int e^{4+x} x^2 \, dx+\frac {2}{9} \int e^{4+x} x \, dx \\ & = -\frac {8}{3} e^{2+\frac {x}{2}}+\frac {2}{9} e^{4+x} x+\frac {1}{9} e^{4+x} x^2+\frac {4}{3} e^{2+\frac {x}{2}} (2+x)-\frac {2}{9} \int e^{4+x} \, dx-\frac {2}{9} \int e^{4+x} x \, dx \\ & = -\frac {8}{3} e^{2+\frac {x}{2}}-\frac {2 e^{4+x}}{9}+\frac {1}{9} e^{4+x} x^2+\frac {4}{3} e^{2+\frac {x}{2}} (2+x)+\frac {2}{9} \int e^{4+x} \, dx \\ & = -\frac {8}{3} e^{2+\frac {x}{2}}+\frac {1}{9} e^{4+x} x^2+\frac {4}{3} e^{2+\frac {x}{2}} (2+x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {1}{9} \left (6+e^{2+\frac {x}{2}} x\right )^2 \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {4 x \,{\mathrm e}^{2+\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4+x}}{9}\) | \(20\) |
derivativedivides | \(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{9}\) | \(26\) |
default | \(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{9}\) | \(26\) |
norman | \(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{9}\) | \(26\) |
parallelrisch | \(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{9}\) | \(26\) |
parts | \(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{2}}}{3}+\frac {x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{9}\) | \(26\) |
meijerg | \(-\frac {{\mathrm e}^{4} \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )}{9}+\frac {2 \,{\mathrm e}^{4} \left (1-\frac {\left (2-2 x \right ) {\mathrm e}^{x}}{2}\right )}{9}-\frac {8 \,{\mathrm e}^{2} \left (1-{\mathrm e}^{\frac {x}{2}}\right )}{3}+\frac {8 \,{\mathrm e}^{2} \left (1-\frac {\left (2-x \right ) {\mathrm e}^{\frac {x}{2}}}{2}\right )}{3}\) | \(66\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {1}{9} \, x^{2} e^{\left (x + 4\right )} + \frac {4}{3} \, x e^{\left (\frac {1}{2} \, x + 2\right )} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {x^{2} e^{4} e^{x}}{9} + \frac {4 x e^{2} e^{\frac {x}{2}}}{3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {1}{9} \, x^{2} e^{\left (x + 4\right )} + \frac {4}{3} \, {\left (x e^{2} - 2 \, e^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {8}{3} \, e^{\left (\frac {1}{2} \, x + 2\right )} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {1}{9} \, x^{2} e^{\left (x + 4\right )} + \frac {4}{3} \, x e^{\left (\frac {1}{2} \, x + 2\right )} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{9} \left (e^{2+\frac {x}{2}} (12+6 x)+e^{4+x} \left (2 x+x^2\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{\frac {x}{2}+2}\,\left (x\,{\mathrm {e}}^{\frac {x}{2}+2}+12\right )}{9} \]
[In]
[Out]