\(\int \frac {e (-75-27 x^2-4 x^3)+e^{e^x} (e (-3-x^2)+e^{1+x} (3 x-x^2-x^3))}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} (9-6 x-5 x^2+2 x^3+x^4)+e^{e^x} (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5)} \, dx\) [1829]

Optimal result

Integrand size = 141, antiderivative size = 27 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {e}{\left (25+e^{e^x}+2 x\right ) \left (1+\frac {-3+x^2}{x}\right )} \] Output:

exp(1)/(1+(x^2-3)/x)/(25+2*x+exp(exp(x)))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {e x}{\left (25+e^{e^x}+2 x\right ) \left (-3+x+x^2\right )} \] Input:

Integrate[(E*(-75 - 27*x^2 - 4*x^3) + E^E^x*(E*(-3 - x^2) + E^(1 + x)*(3*x 
 - x^2 - x^3)))/(5625 - 2850*x - 3689*x^2 + 726*x^3 + 805*x^4 + 108*x^5 + 
4*x^6 + E^(2*E^x)*(9 - 6*x - 5*x^2 + 2*x^3 + x^4) + E^E^x*(450 - 264*x - 2 
74*x^2 + 80*x^3 + 58*x^4 + 4*x^5)),x]
 

Output:

(E*x)/((25 + E^E^x + 2*x)*(-3 + x + x^2))
 

Rubi [F]

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.

\(\displaystyle \int \frac {e \left (-4 x^3-27 x^2-75\right )+e^{e^x} \left (e \left (-x^2-3\right )+e^{x+1} \left (-x^3-x^2+3 x\right )\right )}{4 x^6+108 x^5+805 x^4+726 x^3-3689 x^2+e^{2 e^x} \left (x^4+2 x^3-5 x^2-6 x+9\right )+e^{e^x} \left (4 x^5+58 x^4+80 x^3-274 x^2-264 x+450\right )-2850 x+5625} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e \left (-4 x^3-27 x^2-e^{x+e^x} \left (x^2+x-3\right ) x-e^{e^x} \left (x^2+3\right )-75\right )}{\left (2 x+e^{e^x}+25\right )^2 \left (-x^2-x+3\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle e \int -\frac {4 x^3+27 x^2-e^{x+e^x} \left (-x^2-x+3\right ) x+e^{e^x} \left (x^2+3\right )+75}{\left (2 x+e^{e^x}+25\right )^2 \left (-x^2-x+3\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -e \int \frac {4 x^3+27 x^2-e^{x+e^x} \left (-x^2-x+3\right ) x+e^{e^x} \left (x^2+3\right )+75}{\left (2 x+e^{e^x}+25\right )^2 \left (-x^2-x+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -e \int \left (\frac {4 x^3}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}+\frac {27 x^2}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}+\frac {e^{x+e^x} x}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )}+\frac {e^{e^x} \left (x^2+3\right )}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}+\frac {75}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -e \left (144 \int \frac {1}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}dx+6 \int \frac {e^{e^x}}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}dx-11 \int \frac {x}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}dx-\int \frac {e^{e^x} x}{\left (2 x+e^{e^x}+25\right )^2 \left (x^2+x-3\right )^2}dx-\frac {46 \int \frac {1}{\left (-2 x+\sqrt {13}-1\right ) \left (2 x+e^{e^x}+25\right )^2}dx}{\sqrt {13}}-\frac {2 \int \frac {e^{e^x}}{\left (-2 x+\sqrt {13}-1\right ) \left (2 x+e^{e^x}+25\right )^2}dx}{\sqrt {13}}+\frac {4}{13} \left (13-\sqrt {13}\right ) \int \frac {1}{\left (2 x-\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx+\frac {1}{13} \left (13-\sqrt {13}\right ) \int \frac {e^{x+e^x}}{\left (2 x-\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx+\frac {4}{13} \left (13+\sqrt {13}\right ) \int \frac {1}{\left (2 x+\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx-\frac {46 \int \frac {1}{\left (2 x+\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx}{\sqrt {13}}-\frac {2 \int \frac {e^{e^x}}{\left (2 x+\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx}{\sqrt {13}}+\frac {1}{13} \left (13+\sqrt {13}\right ) \int \frac {e^{x+e^x}}{\left (2 x+\sqrt {13}+1\right ) \left (2 x+e^{e^x}+25\right )^2}dx\right )\)

Input:

Int[(E*(-75 - 27*x^2 - 4*x^3) + E^E^x*(E*(-3 - x^2) + E^(1 + x)*(3*x - x^2 
 - x^3)))/(5625 - 2850*x - 3689*x^2 + 726*x^3 + 805*x^4 + 108*x^5 + 4*x^6 
+ E^(2*E^x)*(9 - 6*x - 5*x^2 + 2*x^3 + x^4) + E^E^x*(450 - 264*x - 274*x^2 
 + 80*x^3 + 58*x^4 + 4*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {x \,{\mathrm e}}{\left (x^{2}+x -3\right ) \left (25+2 x +{\mathrm e}^{{\mathrm e}^{x}}\right )}\)	\(23\)
parallelrisch	\(\frac {x \,{\mathrm e}}{{\mathrm e}^{{\mathrm e}^{x}} x^{2}+2 x^{3}+x \,{\mathrm e}^{{\mathrm e}^{x}}+27 x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{x}}+19 x -75}\)	\(39\)

Input:

int((((-x^3-x^2+3*x)*exp(1)*exp(x)+(-x^2-3)*exp(1))*exp(exp(x))+(-4*x^3-27 
*x^2-75)*exp(1))/((x^4+2*x^3-5*x^2-6*x+9)*exp(exp(x))^2+(4*x^5+58*x^4+80*x 
^3-274*x^2-264*x+450)*exp(exp(x))+4*x^6+108*x^5+805*x^4+726*x^3-3689*x^2-2 
850*x+5625),x,method=_RETURNVERBOSE)
 

Output:

x*exp(1)/(x^2+x-3)/(25+2*x+exp(exp(x)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {x e}{2 \, x^{3} + 27 \, x^{2} + {\left (x^{2} + x - 3\right )} e^{\left (e^{x}\right )} + 19 \, x - 75} \] Input:

integrate((((-x^3-x^2+3*x)*exp(1)*exp(x)+(-x^2-3)*exp(1))*exp(exp(x))+(-4* 
x^3-27*x^2-75)*exp(1))/((x^4+2*x^3-5*x^2-6*x+9)*exp(exp(x))^2+(4*x^5+58*x^ 
4+80*x^3-274*x^2-264*x+450)*exp(exp(x))+4*x^6+108*x^5+805*x^4+726*x^3-3689 
*x^2-2850*x+5625),x, algorithm="fricas")
 

Output:

x*e/(2*x^3 + 27*x^2 + (x^2 + x - 3)*e^(e^x) + 19*x - 75)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {e x}{2 x^{3} + 27 x^{2} + 19 x + \left (x^{2} + x - 3\right ) e^{e^{x}} - 75} \] Input:

integrate((((-x**3-x**2+3*x)*exp(1)*exp(x)+(-x**2-3)*exp(1))*exp(exp(x))+( 
-4*x**3-27*x**2-75)*exp(1))/((x**4+2*x**3-5*x**2-6*x+9)*exp(exp(x))**2+(4* 
x**5+58*x**4+80*x**3-274*x**2-264*x+450)*exp(exp(x))+4*x**6+108*x**5+805*x 
**4+726*x**3-3689*x**2-2850*x+5625),x)
 

Output:

E*x/(2*x**3 + 27*x**2 + 19*x + (x**2 + x - 3)*exp(exp(x)) - 75)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {x e}{2 \, x^{3} + 27 \, x^{2} + {\left (x^{2} + x - 3\right )} e^{\left (e^{x}\right )} + 19 \, x - 75} \] Input:

integrate((((-x^3-x^2+3*x)*exp(1)*exp(x)+(-x^2-3)*exp(1))*exp(exp(x))+(-4* 
x^3-27*x^2-75)*exp(1))/((x^4+2*x^3-5*x^2-6*x+9)*exp(exp(x))^2+(4*x^5+58*x^ 
4+80*x^3-274*x^2-264*x+450)*exp(exp(x))+4*x^6+108*x^5+805*x^4+726*x^3-3689 
*x^2-2850*x+5625),x, algorithm="maxima")
 

Output:

x*e/(2*x^3 + 27*x^2 + (x^2 + x - 3)*e^(e^x) + 19*x - 75)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {x e}{2 \, x^{3} + x^{2} e^{\left (e^{x}\right )} + 27 \, x^{2} + x e^{\left (e^{x}\right )} + 19 \, x - 3 \, e^{\left (e^{x}\right )} - 75} \] Input:

integrate((((-x^3-x^2+3*x)*exp(1)*exp(x)+(-x^2-3)*exp(1))*exp(exp(x))+(-4* 
x^3-27*x^2-75)*exp(1))/((x^4+2*x^3-5*x^2-6*x+9)*exp(exp(x))^2+(4*x^5+58*x^ 
4+80*x^3-274*x^2-264*x+450)*exp(exp(x))+4*x^6+108*x^5+805*x^4+726*x^3-3689 
*x^2-2850*x+5625),x, algorithm="giac")
 

Output:

x*e/(2*x^3 + x^2*e^(e^x) + 27*x^2 + x*e^(e^x) + 19*x - 3*e^(e^x) - 75)
 

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.19 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {2\,x^4\,{\mathrm {e}}^{x+1}-x\,\left (75\,{\mathrm {e}}^{x+1}-6\,\mathrm {e}\right )+x^2\,\left (19\,{\mathrm {e}}^{x+1}-2\,\mathrm {e}\right )+x^3\,\left (27\,{\mathrm {e}}^{x+1}-2\,\mathrm {e}\right )}{\left (25\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x-2\right )\,\left (2\,x+{\mathrm {e}}^{{\mathrm {e}}^x}+25\right )\,{\left (x^2+x-3\right )}^2} \] Input:

int(-(exp(exp(x))*(exp(1)*(x^2 + 3) + exp(1)*exp(x)*(x^2 - 3*x + x^3)) + e 
xp(1)*(27*x^2 + 4*x^3 + 75))/(exp(2*exp(x))*(2*x^3 - 5*x^2 - 6*x + x^4 + 9 
) - 2850*x + exp(exp(x))*(80*x^3 - 274*x^2 - 264*x + 58*x^4 + 4*x^5 + 450) 
 - 3689*x^2 + 726*x^3 + 805*x^4 + 108*x^5 + 4*x^6 + 5625),x)
 

Output:

(2*x^4*exp(x + 1) - x*(75*exp(x + 1) - 6*exp(1)) + x^2*(19*exp(x + 1) - 2* 
exp(1)) + x^3*(27*exp(x + 1) - 2*exp(1)))/((25*exp(x) + 2*x*exp(x) - 2)*(2 
*x + exp(exp(x)) + 25)*(x + x^2 - 3)^2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e \left (-75-27 x^2-4 x^3\right )+e^{e^x} \left (e \left (-3-x^2\right )+e^{1+x} \left (3 x-x^2-x^3\right )\right )}{5625-2850 x-3689 x^2+726 x^3+805 x^4+108 x^5+4 x^6+e^{2 e^x} \left (9-6 x-5 x^2+2 x^3+x^4\right )+e^{e^x} \left (450-264 x-274 x^2+80 x^3+58 x^4+4 x^5\right )} \, dx=\frac {e x}{e^{e^{x}} x^{2}+e^{e^{x}} x -3 e^{e^{x}}+2 x^{3}+27 x^{2}+19 x -75} \] Input:

int((((-x^3-x^2+3*x)*exp(1)*exp(x)+(-x^2-3)*exp(1))*exp(exp(x))+(-4*x^3-27 
*x^2-75)*exp(1))/((x^4+2*x^3-5*x^2-6*x+9)*exp(exp(x))^2+(4*x^5+58*x^4+80*x 
^3-274*x^2-264*x+450)*exp(exp(x))+4*x^6+108*x^5+805*x^4+726*x^3-3689*x^2-2 
850*x+5625),x)
 

Output:

(e*x)/(e**(e**x)*x**2 + e**(e**x)*x - 3*e**(e**x) + 2*x**3 + 27*x**2 + 19* 
x - 75)