\(\int \frac {e^{2 x} (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} (27+6831 x+1215 x^2-19 x^3+2 x^4))}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 (2304+768 x+64 x^2)}{9-6 x+x^2}} (-27 x^2+27 x^3-9 x^4+x^5)+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} (162 x-216 x^2+108 x^3-24 x^4+2 x^5)} \, dx\) [1247]

Optimal result

Integrand size = 187, antiderivative size = 34 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{2 x}}{-3+x+e^{16 \left (3+\frac {3+5 x}{3-x}\right )^2} x} \] Output:

exp(x)^2/(x+x*exp(4*(3+(3+5*x)/(3-x))*(12+4*(3+5*x)/(3-x)))-3)
 

Mathematica [A] (verified)

Time = 5.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{2 x}}{-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x} \] Input:

Integrate[(E^(2*x)*(189 - 243*x + 117*x^2 - 25*x^3 + 2*x^4 + E^((2304 + 76 
8*x + 64*x^2)/(9 - 6*x + x^2))*(27 + 6831*x + 1215*x^2 - 19*x^3 + 2*x^4))) 
/(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 + E^((2*(2304 + 768*x + 6 
4*x^2))/(9 - 6*x + x^2))*(-27*x^2 + 27*x^3 - 9*x^4 + x^5) + E^((2304 + 768 
*x + 64*x^2)/(9 - 6*x + x^2))*(162*x - 216*x^2 + 108*x^3 - 24*x^4 + 2*x^5) 
),x]
 

Output:

E^(2*x)/(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)
 

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.

\(\displaystyle \int \frac {e^{2 x} \left (2 x^4-25 x^3+117 x^2+e^{\frac {64 x^2+768 x+2304}{x^2-6 x+9}} \left (2 x^4-19 x^3+1215 x^2+6831 x+27\right )-243 x+189\right )}{x^5-15 x^4+90 x^3-270 x^2+e^{\frac {2 \left (64 x^2+768 x+2304\right )}{x^2-6 x+9}} \left (x^5-9 x^4+27 x^3-27 x^2\right )+e^{\frac {64 x^2+768 x+2304}{x^2-6 x+9}} \left (2 x^5-24 x^4+108 x^3-216 x^2+162 x\right )+405 x-243} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{2 x} \left (-e^{\frac {64 (x+6)^2}{(x-3)^2}} \left (2 x^4-19 x^3+1215 x^2+6831 x+27\right )-\left ((2 x-7) (x-3)^3\right )\right )}{(3-x)^3 \left (-e^{\frac {64 (x+6)^2}{(x-3)^2}} x-x+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{2 x} \left (2 x^4-19 x^3+1215 x^2+6831 x+27\right )}{(x-3)^3 x \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )}-\frac {3 e^{2 x} \left (385 x^2+2298 x+9\right )}{(x-3)^2 x \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -10368 \int \frac {e^{2 x}}{(x-3)^2 \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )^2}dx-1152 \int \frac {e^{2 x}}{(x-3) \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )^2}dx-3 \int \frac {e^{2 x}}{x \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )^2}dx+2 \int \frac {e^{2 x}}{e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3}dx+10368 \int \frac {e^{2 x}}{(x-3)^3 \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )}dx+1152 \int \frac {e^{2 x}}{(x-3)^2 \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )}dx-\int \frac {e^{2 x}}{x \left (e^{\frac {64 (x+6)^2}{(x-3)^2}} x+x-3\right )}dx\)

Input:

Int[(E^(2*x)*(189 - 243*x + 117*x^2 - 25*x^3 + 2*x^4 + E^((2304 + 768*x + 
64*x^2)/(9 - 6*x + x^2))*(27 + 6831*x + 1215*x^2 - 19*x^3 + 2*x^4)))/(-243 
 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 + E^((2*(2304 + 768*x + 64*x^2) 
)/(9 - 6*x + x^2))*(-27*x^2 + 27*x^3 - 9*x^4 + x^5) + E^((2304 + 768*x + 6 
4*x^2)/(9 - 6*x + x^2))*(162*x - 216*x^2 + 108*x^3 - 24*x^4 + 2*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {{\mathrm e}^{2 x}}{x \,{\mathrm e}^{\frac {64 \left (6+x \right )^{2}}{\left (-3+x \right )^{2}}}+x -3}\)	\(26\)
parallelrisch	\(\frac {{\mathrm e}^{2 x}}{x \,{\mathrm e}^{\frac {64 x^{2}+768 x +2304}{x^{2}-6 x +9}}+x -3}\)	\(34\)

Input:

int(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9) 
)+2*x^4-25*x^3+117*x^2-243*x+189)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp( 
(64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162*x)*ex 
p((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, 
method=_RETURNVERBOSE)
 

Output:

exp(2*x)/(x*exp(64*(6+x)^2/(-3+x)^2)+x-3)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{\left (2 \, x\right )}}{x e^{\left (\frac {64 \, {\left (x^{2} + 12 \, x + 36\right )}}{x^{2} - 6 \, x + 9}\right )} + x - 3} \] Input:

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2- 
6*x+9))+2*x^4-25*x^3+117*x^2-243*x+189)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2 
)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162 
*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-2 
43),x, algorithm="fricas")
 

Output:

e^(2*x)/(x*e^(64*(x^2 + 12*x + 36)/(x^2 - 6*x + 9)) + x - 3)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{2 x}}{x e^{\frac {64 x^{2} + 768 x + 2304}{x^{2} - 6 x + 9}} + x - 3} \] Input:

integrate(((2*x**4-19*x**3+1215*x**2+6831*x+27)*exp((64*x**2+768*x+2304)/( 
x**2-6*x+9))+2*x**4-25*x**3+117*x**2-243*x+189)*exp(x)**2/((x**5-9*x**4+27 
*x**3-27*x**2)*exp((64*x**2+768*x+2304)/(x**2-6*x+9))**2+(2*x**5-24*x**4+1 
08*x**3-216*x**2+162*x)*exp((64*x**2+768*x+2304)/(x**2-6*x+9))+x**5-15*x** 
4+90*x**3-270*x**2+405*x-243),x)
 

Output:

exp(2*x)/(x*exp((64*x**2 + 768*x + 2304)/(x**2 - 6*x + 9)) + x - 3)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{\left (2 \, x\right )}}{x e^{\left (\frac {5184}{x^{2} - 6 \, x + 9} + \frac {1152}{x - 3} + 64\right )} + x - 3} \] Input:

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2- 
6*x+9))+2*x^4-25*x^3+117*x^2-243*x+189)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2 
)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162 
*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-2 
43),x, algorithm="maxima")
 

Output:

e^(2*x)/(x*e^(5184/(x^2 - 6*x + 9) + 1152/(x - 3) + 64) + x - 3)
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{\left (2 \, x\right )}}{x e^{\left (-\frac {192 \, {\left (x^{2} - 12 \, x\right )}}{x^{2} - 6 \, x + 9} + 256\right )} + x - 3} \] Input:

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2- 
6*x+9))+2*x^4-25*x^3+117*x^2-243*x+189)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2 
)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162 
*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-2 
43),x, algorithm="giac")
 

Output:

e^(2*x)/(x*e^(-192*(x^2 - 12*x)/(x^2 - 6*x + 9) + 256) + x - 3)
 

Mupad [B] (verification not implemented)

Time = 7.73 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {{\mathrm {e}}^{2\,x}\,\left (385\,x^2+2298\,x+9\right )\,{\left (x^3-9\,x^2+27\,x-27\right )}^2}{{\left (x-3\right )}^2\,\left (x+x\,{\mathrm {e}}^{\frac {768\,x}{x^2-6\,x+9}+\frac {2304}{x^2-6\,x+9}+\frac {64\,x^2}{x^2-6\,x+9}}-3\right )\,\left (385\,x^6-2322\,x^5-6777\,x^4+82404\,x^3-216513\,x^2+185166\,x+729\right )} \] Input:

int((exp(2*x)*(117*x^2 - 243*x - 25*x^3 + 2*x^4 + exp((768*x + 64*x^2 + 23 
04)/(x^2 - 6*x + 9))*(6831*x + 1215*x^2 - 19*x^3 + 2*x^4 + 27) + 189))/(40 
5*x + exp((768*x + 64*x^2 + 2304)/(x^2 - 6*x + 9))*(162*x - 216*x^2 + 108* 
x^3 - 24*x^4 + 2*x^5) - exp((2*(768*x + 64*x^2 + 2304))/(x^2 - 6*x + 9))*( 
27*x^2 - 27*x^3 + 9*x^4 - x^5) - 270*x^2 + 90*x^3 - 15*x^4 + x^5 - 243),x)
 

Output:

(exp(2*x)*(2298*x + 385*x^2 + 9)*(27*x - 9*x^2 + x^3 - 27)^2)/((x - 3)^2*( 
x + x*exp((768*x)/(x^2 - 6*x + 9) + 2304/(x^2 - 6*x + 9) + (64*x^2)/(x^2 - 
 6*x + 9)) - 3)*(185166*x - 216513*x^2 + 82404*x^3 - 6777*x^4 - 2322*x^5 + 
 385*x^6 + 729))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx=\frac {e^{2 x}}{e^{\frac {1152 x +1728}{x^{2}-6 x +9}} e^{64} x +x -3} \] Input:

int(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9) 
)+2*x^4-25*x^3+117*x^2-243*x+189)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp( 
(64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162*x)*ex 
p((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x)
 

Output:

e**(2*x)/(e**((1152*x + 1728)/(x**2 - 6*x + 9))*e**64*x + x - 3)