Integrand size = 113, antiderivative size = 29 \[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=e^{\frac {e^{2 x-\frac {3 x}{2+x \left (20+\log \left (e^{2 x}\right )\right )}}}{x}} \] Output:
exp(exp(2*x-3*x/(2+x*(20+ln(exp(x)^2))))/x)
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=e^{\frac {e^{2 x-\frac {3 x}{2 \left (1+10 x+x^2\right )}}}{x}} \] Input:
Integrate[(E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))*(-2 - 39*x - 124*x^2 + 371*x^3 + 78*x^4 + 4*x ^5))/(2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x]
Output:
E^(E^(2*x - (3*x)/(2*(1 + 10*x + x^2)))/x)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (4 x^5+78 x^4+371 x^3-124 x^2-39 x-2\right ) \exp \left (\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}}}{x}\right )}{2 x^6+40 x^5+204 x^4+40 x^3+2 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (4 x^5+78 x^4+371 x^3-124 x^2-39 x-2\right ) \exp \left (\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}}}{x}\right )}{x^2 \left (2 x^4+40 x^3+204 x^2+40 x+2\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {\left (4 x^5+78 x^4+371 x^3-124 x^2-39 x-2\right ) \exp \left (\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 x^2+20 x+2}}}{x}\right )}{2 x^2 \left (x^2+10 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right ) \left (-4 x^5-78 x^4-371 x^3+124 x^2+39 x+2\right )}{x^2 \left (x^2+10 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right ) \left (-4 x^5-78 x^4-371 x^3+124 x^2+39 x+2\right )}{x^2 \left (x^2+10 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (-\frac {6 \exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right ) (x+5)}{\left (x^2+10 x+1\right )^2}-\frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{x}-\frac {3 \exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right ) (x+10)}{x^2+10 x+1}+\frac {2 \exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{4} \left (5-2 \sqrt {6}\right ) \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{\left (-2 x+4 \sqrt {6}-10\right )^2}dx+\frac {5}{4} \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{\left (-2 x+4 \sqrt {6}-10\right )^2}dx-2 \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{x^2}dx+\int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{x}dx+\frac {1}{4} \left (12+5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{2 x-4 \sqrt {6}+10}dx-\frac {1}{4} \left (5+2 \sqrt {6}\right ) \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{\left (2 x+4 \sqrt {6}+10\right )^2}dx+\frac {5}{4} \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{\left (2 x+4 \sqrt {6}+10\right )^2}dx+\frac {1}{4} \left (12-5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}+\frac {e^{\frac {4 x^3+40 x^2+x}{2 \left (x^2+10 x+1\right )}}}{x}\right )}{2 x+4 \sqrt {6}+10}dx\right )\) |
Input:
Int[(E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^ 3)/(2 + 20*x + 2*x^2))*(-2 - 39*x - 124*x^2 + 371*x^3 + 78*x^4 + 4*x^5))/( 2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x]
Output:
$Aborted
Time = 3.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \({\mathrm e}^{\frac {{\mathrm e}^{\frac {x \left (4 x^{2}+40 x +1\right )}{2 x^{2}+20 x +2}}}{x}}\) | \(30\) |
| parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{\frac {x \left (4 x^{2}+40 x +1\right )}{2 x^{2}+20 x +2}}}{x}}\) | \(30\) |
| norman | \(\frac {x \,{\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}+x^{3} {\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}+10 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}}{x \left (x^{2}+10 x +1\right )}\) | \(120\) |
Input:
int((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x +2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40* x^3+2*x^2),x,method=_RETURNVERBOSE)
Output:
exp(exp(1/2*x*(4*x^2+40*x+1)/(x^2+10*x+1))/x)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.07 \[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=e^{\left (\frac {4 \, x^{4} + 40 \, x^{3} + x^{2} + 2 \, {\left (x^{2} + 10 \, x + 1\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{2 \, {\left (x^{3} + 10 \, x^{2} + x\right )}} - \frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )} \] Input:
integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^ 2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x ^4+40*x^3+2*x^2),x, algorithm="fricas")
Output:
e^(1/2*(4*x^4 + 40*x^3 + x^2 + 2*(x^2 + 10*x + 1)*e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1)))/(x^3 + 10*x^2 + x) - 1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=e^{\frac {e^{\frac {4 x^{3} + 40 x^{2} + x}{2 x^{2} + 20 x + 2}}}{x}} \] Input:
integrate((4*x**5+78*x**4+371*x**3-124*x**2-39*x-2)*exp((4*x**3+40*x**2+x) /(2*x**2+20*x+2))*exp(exp((4*x**3+40*x**2+x)/(2*x**2+20*x+2))/x)/(2*x**6+4 0*x**5+204*x**4+40*x**3+2*x**2),x)
Output:
exp(exp((4*x**3 + 40*x**2 + x)/(2*x**2 + 20*x + 2))/x)
\[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=\int { \frac {{\left (4 \, x^{5} + 78 \, x^{4} + 371 \, x^{3} - 124 \, x^{2} - 39 \, x - 2\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}} + \frac {e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{x}\right )}}{2 \, {\left (x^{6} + 20 \, x^{5} + 102 \, x^{4} + 20 \, x^{3} + x^{2}\right )}} \,d x } \] Input:
integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^ 2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x ^4+40*x^3+2*x^2),x, algorithm="maxima")
Output:
1/2*integrate((4*x^5 + 78*x^4 + 371*x^3 - 124*x^2 - 39*x - 2)*e^(1/2*(4*x^ 3 + 40*x^2 + x)/(x^2 + 10*x + 1) + e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))/x)/(x^6 + 20*x^5 + 102*x^4 + 20*x^3 + x^2), x)
\[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=\int { \frac {{\left (4 \, x^{5} + 78 \, x^{4} + 371 \, x^{3} - 124 \, x^{2} - 39 \, x - 2\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}} + \frac {e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{x}\right )}}{2 \, {\left (x^{6} + 20 \, x^{5} + 102 \, x^{4} + 20 \, x^{3} + x^{2}\right )}} \,d x } \] Input:
integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^ 2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x ^4+40*x^3+2*x^2),x, algorithm="giac")
Output:
integrate(1/2*(4*x^5 + 78*x^4 + 371*x^3 - 124*x^2 - 39*x - 2)*e^(1/2*(4*x^ 3 + 40*x^2 + x)/(x^2 + 10*x + 1) + e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))/x)/(x^6 + 20*x^5 + 102*x^4 + 20*x^3 + x^2), x)
Time = 3.79 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {x}{2\,x^2+20\,x+2}}\,{\mathrm {e}}^{\frac {2\,x^3}{x^2+10\,x+1}}\,{\mathrm {e}}^{\frac {20\,x^2}{x^2+10\,x+1}}}{x}} \] Input:
int(-(exp(exp((x + 40*x^2 + 4*x^3)/(20*x + 2*x^2 + 2))/x)*exp((x + 40*x^2 + 4*x^3)/(20*x + 2*x^2 + 2))*(39*x + 124*x^2 - 371*x^3 - 78*x^4 - 4*x^5 + 2))/(2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x)
Output:
exp((exp(x/(20*x + 2*x^2 + 2))*exp((2*x^3)/(10*x + x^2 + 1))*exp((20*x^2)/ (10*x + x^2 + 1)))/x)
\[ \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx=-\left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{6}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{5}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}}d x \right )-\frac {39 \left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{5}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{\frac {3 x}{2 x^{2}+20 x +2}} x}d x \right )}{2}-62 \left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x +e^{\frac {3 x}{2 x^{2}+20 x +2}}}d x \right )+2 \left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}} x^{3}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x +e^{\frac {3 x}{2 x^{2}+20 x +2}}}d x \right )+39 \left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}} x^{2}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x +e^{\frac {3 x}{2 x^{2}+20 x +2}}}d x \right )+\frac {371 \left (\int \frac {e^{\frac {2 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+e^{2 x}}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x}} x}{e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{4}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{3}+102 e^{\frac {3 x}{2 x^{2}+20 x +2}} x^{2}+20 e^{\frac {3 x}{2 x^{2}+20 x +2}} x +e^{\frac {3 x}{2 x^{2}+20 x +2}}}d x \right )}{2} \] Input:
int((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x +2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40* x^3+2*x^2),x)
Output:
( - 2*int(e**((2*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e**(2*x))/(e**((3*x )/(2*x**2 + 20*x + 2))*x))/(e**((3*x)/(2*x**2 + 20*x + 2))*x**6 + 20*e**(( 3*x)/(2*x**2 + 20*x + 2))*x**5 + 102*e**((3*x)/(2*x**2 + 20*x + 2))*x**4 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x**3 + e**((3*x)/(2*x**2 + 20*x + 2))*x **2),x) - 39*int(e**((2*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e**(2*x))/(e **((3*x)/(2*x**2 + 20*x + 2))*x))/(e**((3*x)/(2*x**2 + 20*x + 2))*x**5 + 2 0*e**((3*x)/(2*x**2 + 20*x + 2))*x**4 + 102*e**((3*x)/(2*x**2 + 20*x + 2)) *x**3 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e**((3*x)/(2*x**2 + 20*x + 2))*x),x) - 124*int(e**((2*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e**(2*x ))/(e**((3*x)/(2*x**2 + 20*x + 2))*x))/(e**((3*x)/(2*x**2 + 20*x + 2))*x** 4 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x**3 + 102*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x + e**((3*x)/(2*x**2 + 20* x + 2))),x) + 4*int((e**((2*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e**(2*x) )/(e**((3*x)/(2*x**2 + 20*x + 2))*x))*x**3)/(e**((3*x)/(2*x**2 + 20*x + 2) )*x**4 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x**3 + 102*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x + e**((3*x)/(2*x**2 + 20*x + 2))),x) + 78*int((e**((2*e**((3*x)/(2*x**2 + 20*x + 2))*x**2 + e* *(2*x))/(e**((3*x)/(2*x**2 + 20*x + 2))*x))*x**2)/(e**((3*x)/(2*x**2 + 20* x + 2))*x**4 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x**3 + 102*e**((3*x)/(2*x **2 + 20*x + 2))*x**2 + 20*e**((3*x)/(2*x**2 + 20*x + 2))*x + e**((3*x)...