\(\int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} (24 x+24 x^2+6 x^3-2 x^2 \log (25))))}{4+4 x+x^2} \, dx\) [1106]

Optimal result

Integrand size = 182, antiderivative size = 28 \[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x \left (2+e^{3 e^{\left (25^{\frac {1}{2+\frac {x}{3+x}}}+x\right )^2}}+x\right ) \] Output:

(2+exp(3*exp((exp(2*ln(5)/(x/(3+x)+2))+x)^2))+x)*x
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x \left (2+e^{3 e^{\left (5^{\frac {2 (3+x)}{3 (2+x)}}+x\right )^2}}+x\right ) \] Input:

Integrate[(8 + 16*x + 10*x^2 + 2*x^3 + E^(3*E^(25^((2*(3 + x))/(6 + 3*x)) 
+ 2*25^((3 + x)/(6 + 3*x))*x + x^2))*(4 + 4*x + x^2 + E^(25^((2*(3 + x))/( 
6 + 3*x)) + 2*25^((3 + x)/(6 + 3*x))*x + x^2)*(24*x^2 + 24*x^3 + 6*x^4 - 2 
*25^((2*(3 + x))/(6 + 3*x))*x*Log[25] + 25^((3 + x)/(6 + 3*x))*(24*x + 24* 
x^2 + 6*x^3 - 2*x^2*Log[25]))))/(4 + 4*x + x^2),x]
 

Output:

x*(2 + E^(3*E^(5^((2*(3 + x))/(3*(2 + x))) + 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.

\(\displaystyle \int \frac {\exp \left (3 \exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right )\right ) \left (\exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right ) \left (6 x^4+24 x^3+24 x^2+25^{\frac {x+3}{3 x+6}} \left (6 x^3+24 x^2-2 x^2 \log (25)+24 x\right )-2\ 25^{\frac {2 (x+3)}{3 x+6}} x \log (25)\right )+x^2+4 x+4\right )+2 x^3+10 x^2+16 x+8}{x^2+4 x+4} \, dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {\exp \left (3 \exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right )\right ) \left (\exp \left (x^2+2\ 25^{\frac {x+3}{3 x+6}} x+25^{\frac {2 (x+3)}{3 x+6}}\right ) \left (6 x^4+24 x^3+24 x^2+25^{\frac {x+3}{3 x+6}} \left (6 x^3+24 x^2-2 x^2 \log (25)+24 x\right )-2\ 25^{\frac {2 (x+3)}{3 x+6}} x \log (25)\right )+x^2+4 x+4\right )+2 x^3+10 x^2+16 x+8}{(x+2)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right ) x \exp \left (\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2+3 e^{\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2}\right ) \left (3 x^2+12 x-5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}} \log (25)+12\right )}{(x+2)^2}+\frac {2 x^3}{(x+2)^2}+\frac {e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x^2}{(x+2)^2}+\frac {10 x^2}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x}{(x+2)^2}+\frac {16 x}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}}}{(x+2)^2}+\frac {8}{(x+2)^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {2 \left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right ) x \exp \left (\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2+3 e^{\left (x+5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}}\right )^2}\right ) \left (3 x^2+12 x-5^{\frac {2 x}{3 (x+2)}+\frac {2}{x+2}} \log (25)+12\right )}{(x+2)^2}+\frac {2 x^3}{(x+2)^2}+\frac {e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x^2}{(x+2)^2}+\frac {10 x^2}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}} x}{(x+2)^2}+\frac {16 x}{(x+2)^2}+\frac {4 e^{3 e^{\left (x+25^{\frac {x}{3 x+6}+\frac {3}{3 x+6}}\right )^2}}}{(x+2)^2}+\frac {8}{(x+2)^2}\right )dx\)

Input:

Int[(8 + 16*x + 10*x^2 + 2*x^3 + E^(3*E^(25^((2*(3 + x))/(6 + 3*x)) + 2*25 
^((3 + x)/(6 + 3*x))*x + x^2))*(4 + 4*x + x^2 + E^(25^((2*(3 + x))/(6 + 3* 
x)) + 2*25^((3 + x)/(6 + 3*x))*x + x^2)*(24*x^2 + 24*x^3 + 6*x^4 - 2*25^(( 
2*(3 + x))/(6 + 3*x))*x*Log[25] + 25^((3 + x)/(6 + 3*x))*(24*x + 24*x^2 + 
6*x^3 - 2*x^2*Log[25]))))/(4 + 4*x + x^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 16.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
risch	\(x^{2}+x \,{\mathrm e}^{3 \,{\mathrm e}^{\left (5^{\frac {2+\frac {2 x}{3}}{2+x}}+x \right )^{2}}}+2 x\)	\(30\)
parallelrisch	\(x^{2}+{\mathrm e}^{3 \,{\mathrm e}^{{\mathrm e}^{\frac {4 \left (3+x \right ) \ln \left (5\right )}{3 \left (2+x \right )}}+2 x \,{\mathrm e}^{\frac {2 \left (3+x \right ) \ln \left (5\right )}{3 \left (2+x \right )}}+x^{2}}} x +2 x -40\)	\(50\)

Input:

int((((-4*x*ln(5)*exp(2*(3+x)*ln(5)/(6+3*x))^2+(-4*x^2*ln(5)+6*x^3+24*x^2+ 
24*x)*exp(2*(3+x)*ln(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2*(3+x)*ln(5 
)/(6+3*x))^2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x^2)+x^2+4*x+4)*exp(3*exp(exp( 
2*(3+x)*ln(5)/(6+3*x))^2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x^2))+2*x^3+10*x^2 
+16*x+8)/(x^2+4*x+4),x,method=_RETURNVERBOSE)
 

Output:

x^2+x*exp(3*exp((5^(2/3*(3+x)/(2+x))+x)^2))+2*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x^{2} + x e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 2 \, x \] Input:

integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 
3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* 
(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex 
p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) 
)+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm="fricas")
 

Output:

x^2 + x*e^(3*e^(2*5^(2/3*(x + 3)/(x + 2))*x + x^2 + 5^(4/3*(x + 3)/(x + 2) 
))) + 2*x
 

Sympy [F(-1)]

Timed out. \[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\text {Timed out} \] Input:

integrate((((-4*x*ln(5)*exp(2*(3+x)*ln(5)/(6+3*x))**2+(-4*x**2*ln(5)+6*x** 
3+24*x**2+24*x)*exp(2*(3+x)*ln(5)/(6+3*x))+6*x**4+24*x**3+24*x**2)*exp(exp 
(2*(3+x)*ln(5)/(6+3*x))**2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x**2)+x**2+4*x+4 
)*exp(3*exp(exp(2*(3+x)*ln(5)/(6+3*x))**2+2*x*exp(2*(3+x)*ln(5)/(6+3*x))+x 
**2))+2*x**3+10*x**2+16*x+8)/(x**2+4*x+4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\int { \frac {2 \, x^{3} + 10 \, x^{2} + {\left (x^{2} + 2 \, {\left (3 \, x^{4} + 12 \, x^{3} - 2 \cdot 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x \log \left (5\right ) + {\left (3 \, x^{3} - 2 \, x^{2} \log \left (5\right ) + 12 \, x^{2} + 12 \, x\right )} 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} + 12 \, x^{2}\right )} e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )} + 4 \, x + 4\right )} e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 16 \, x + 8}{x^{2} + 4 \, x + 4} \,d x } \] Input:

integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 
3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* 
(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex 
p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) 
)+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm="maxima")
 

Output:

x^2 + x*e^(3*e^(2*5^(2/3)*5^(2/3/(x + 2))*x + x^2 + 5*5^(1/3)*5^(4/3/(x + 
2)))) + 2*x - integrate(0, x)
 

Giac [F]

\[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\int { \frac {2 \, x^{3} + 10 \, x^{2} + {\left (x^{2} + 2 \, {\left (3 \, x^{4} + 12 \, x^{3} - 2 \cdot 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x \log \left (5\right ) + {\left (3 \, x^{3} - 2 \, x^{2} \log \left (5\right ) + 12 \, x^{2} + 12 \, x\right )} 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} + 12 \, x^{2}\right )} e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )} + 4 \, x + 4\right )} e^{\left (3 \, e^{\left (2 \cdot 5^{\frac {2 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}} x + x^{2} + 5^{\frac {4 \, {\left (x + 3\right )}}{3 \, {\left (x + 2\right )}}}\right )}\right )} + 16 \, x + 8}{x^{2} + 4 \, x + 4} \,d x } \] Input:

integrate((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^ 
3+24*x^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2* 
(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*ex 
p(3*exp(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2) 
)+2*x^3+10*x^2+16*x+8)/(x^2+4*x+4),x, algorithm="giac")
 

Output:

integrate((2*x^3 + 10*x^2 + (x^2 + 2*(3*x^4 + 12*x^3 - 2*5^(4/3*(x + 3)/(x 
 + 2))*x*log(5) + (3*x^3 - 2*x^2*log(5) + 12*x^2 + 12*x)*5^(2/3*(x + 3)/(x 
 + 2)) + 12*x^2)*e^(2*5^(2/3*(x + 3)/(x + 2))*x + x^2 + 5^(4/3*(x + 3)/(x 
+ 2))) + 4*x + 4)*e^(3*e^(2*5^(2/3*(x + 3)/(x + 2))*x + x^2 + 5^(4/3*(x + 
3)/(x + 2)))) + 16*x + 8)/(x^2 + 4*x + 4), x)
 

Mupad [B] (verification not implemented)

Time = 7.82 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=x\,\left (x+{\mathrm {e}}^{3\,{\mathrm {e}}^{2\,5^{\frac {2\,\left (x+3\right )}{3\,\left (x+2\right )}}\,x}\,{\mathrm {e}}^{5^{\frac {4\,\left (x+3\right )}{3\,\left (x+2\right )}}}\,{\mathrm {e}}^{x^2}}+2\right ) \] Input:

int((16*x + exp(3*exp(exp((4*log(5)*(x + 3))/(3*x + 6)) + 2*x*exp((2*log(5 
)*(x + 3))/(3*x + 6)) + x^2))*(4*x + exp(exp((4*log(5)*(x + 3))/(3*x + 6)) 
 + 2*x*exp((2*log(5)*(x + 3))/(3*x + 6)) + x^2)*(exp((2*log(5)*(x + 3))/(3 
*x + 6))*(24*x - 4*x^2*log(5) + 24*x^2 + 6*x^3) + 24*x^2 + 24*x^3 + 6*x^4 
- 4*x*exp((4*log(5)*(x + 3))/(3*x + 6))*log(5)) + x^2 + 4) + 10*x^2 + 2*x^ 
3 + 8)/(4*x + x^2 + 4),x)
 

Output:

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

Reduce [F]

\[ \int \frac {8+16 x+10 x^2+2 x^3+e^{3 e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2}} \left (4+4 x+x^2+e^{25^{\frac {2 (3+x)}{6+3 x}}+2\ 25^{\frac {3+x}{6+3 x}} x+x^2} \left (24 x^2+24 x^3+6 x^4-2\ 25^{\frac {2 (3+x)}{6+3 x}} x \log (25)+25^{\frac {3+x}{6+3 x}} \left (24 x+24 x^2+6 x^3-2 x^2 \log (25)\right )\right )\right )}{4+4 x+x^2} \, dx=\int \frac {\left (\left (-4 x \,\mathrm {log}\left (5\right ) \left ({\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}\right )^{2}+\left (-4 \,\mathrm {log}\left (5\right ) x^{2}+6 x^{3}+24 x^{2}+24 x \right ) {\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}+6 x^{4}+24 x^{3}+24 x^{2}\right ) {\mathrm e}^{\left ({\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}\right )^{2}+2 x \,{\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}+x^{2}}+x^{2}+4 x +4\right ) {\mathrm e}^{3 \,{\mathrm e}^{\left ({\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}\right )^{2}+2 x \,{\mathrm e}^{\frac {2 \left (x +3\right ) \mathrm {log}\left (5\right )}{6+3 x}}+x^{2}}}+2 x^{3}+10 x^{2}+16 x +8}{x^{2}+4 x +4}d x \] Input:

int((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^3+24*x 
^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2*(3+x)* 
log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*exp(3*ex 
p(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2))+2*x^ 
3+10*x^2+16*x+8)/(x^2+4*x+4),x)
 

Output:

int((((-4*x*log(5)*exp(2*(3+x)*log(5)/(6+3*x))^2+(-4*x^2*log(5)+6*x^3+24*x 
^2+24*x)*exp(2*(3+x)*log(5)/(6+3*x))+6*x^4+24*x^3+24*x^2)*exp(exp(2*(3+x)* 
log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2)+x^2+4*x+4)*exp(3*ex 
p(exp(2*(3+x)*log(5)/(6+3*x))^2+2*x*exp(2*(3+x)*log(5)/(6+3*x))+x^2))+2*x^ 
3+10*x^2+16*x+8)/(x^2+4*x+4),x)