\(\int \frac {(50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 (450 x^2-300 x^3+50 x^4)+e^x (-300 x+300 x^2-50 x^3+e^5 (-300 x+100 x^2))) \log (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 (-6 x^2+2 x^3)}{e^x-3 x+x^2})}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 (9 x^3-6 x^4+x^5)+e^x (-3 x^2+x^3+e^5 (-6 x^2+2 x^3))} \, dx\) [558]

Optimal result

Integrand size = 196, antiderivative size = 32 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=e+25 \log ^2\left (2 x \left (e^5+\frac {x}{-\frac {e^x}{3-x}+x}\right )\right ) \] Output:

exp(1)+25*ln(2*(exp(5)+x/(x-exp(x)/(3-x)))*x)^2
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \log ^2\left (\frac {2 x \left (e^{5+x}+(-3+x) x+e^5 (-3+x) x\right )}{e^x+(-3+x) x}\right ) \] Input:

Integrate[((50*E^(5 + 2*x) + 450*x^2 - 300*x^3 + 50*x^4 + E^5*(450*x^2 - 3 
00*x^3 + 50*x^4) + E^x*(-300*x + 300*x^2 - 50*x^3 + E^5*(-300*x + 100*x^2) 
))*Log[(2*E^(5 + x)*x - 6*x^2 + 2*x^3 + E^5*(-6*x^2 + 2*x^3))/(E^x - 3*x + 
 x^2)])/(E^(5 + 2*x)*x + 9*x^3 - 6*x^4 + x^5 + E^5*(9*x^3 - 6*x^4 + x^5) + 
 E^x*(-3*x^2 + x^3 + E^5*(-6*x^2 + 2*x^3))),x]
 

Output:

25*Log[(2*x*(E^(5 + x) + (-3 + x)*x + E^5*(-3 + x)*x))/(E^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.

\(\displaystyle \int \frac {\left (50 x^4-300 x^3+450 x^2+e^x \left (-50 x^3+300 x^2+e^5 \left (100 x^2-300 x\right )-300 x\right )+e^5 \left (50 x^4-300 x^3+450 x^2\right )+50 e^{2 x+5}\right ) \log \left (\frac {2 x^3-6 x^2+e^5 \left (2 x^3-6 x^2\right )+2 e^{x+5} x}{x^2-3 x+e^x}\right )}{x^5-6 x^4+9 x^3+e^x \left (x^3-3 x^2+e^5 \left (2 x^3-6 x^2\right )\right )+e^5 \left (x^5-6 x^4+9 x^3\right )+e^{2 x+5} x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {50 \left (\left (1+e^5\right ) (x-3)^2 x^2-e^x \left (x^2-6 x+6\right ) x+2 e^{x+5} (x-3) x+e^{2 x+5}\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x \left ((x-3) x+e^x\right ) \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 50 \int \frac {\left (\left (1+e^5\right ) (3-x)^2 x^2-2 e^{x+5} (3-x) x-e^x \left (x^2-6 x+6\right ) x+e^{2 x+5}\right ) \log \left (\frac {2 x \left (e^{x+5}-\left (1+e^5\right ) (3-x) x\right )}{e^x-(3-x) x}\right )}{x \left (e^x-(3-x) x\right ) \left (e^{x+5}-\left (1+e^5\right ) (3-x) x\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 50 \int \left (\frac {\log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x}+\frac {\left (x^2-5 x+3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x^2-3 x+e^x}+\frac {\left (1+e^5\right ) \left (-x^2+5 x-3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{\left (1+e^5\right ) x^2-3 \left (1+e^5\right ) x+e^{x+5}}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle 50 \int \left (\frac {\log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x}+\frac {\left (x^2-5 x+3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x^2-3 x+e^x}+\frac {\left (1+e^5\right ) \left (-x^2+5 x-3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{\left (1+e^5\right ) x^2-3 \left (1+e^5\right ) x+e^{x+5}}\right )dx\)

Input:

Int[((50*E^(5 + 2*x) + 450*x^2 - 300*x^3 + 50*x^4 + E^5*(450*x^2 - 300*x^3 
 + 50*x^4) + E^x*(-300*x + 300*x^2 - 50*x^3 + E^5*(-300*x + 100*x^2)))*Log 
[(2*E^(5 + x)*x - 6*x^2 + 2*x^3 + E^5*(-6*x^2 + 2*x^3))/(E^x - 3*x + x^2)] 
)/(E^(5 + 2*x)*x + 9*x^3 - 6*x^4 + x^5 + E^5*(9*x^3 - 6*x^4 + x^5) + E^x*( 
-3*x^2 + x^3 + E^5*(-6*x^2 + 2*x^3))),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.46 (sec) , antiderivative size = 1829, normalized size of antiderivative = 57.16

method	result	size
risch	\(\text {Expression too large to display}\)	\(1829\)

Input:

int((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x)*exp( 
x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*ln((2*x*exp(5)* 
exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5)*exp(x 
)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5)+x^5-6 
*x^4+9*x^3),x,method=_RETURNVERBOSE)
 

Output:

-25*I*Pi*ln(x^2-3*x+exp(-5)*x^2-3*exp(-5)*x+exp(x))*csgn(I*((exp(x)+x^2-3* 
x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^ 
2-3*x)*exp(5)+x^2-3*x))*csgn(I*x)+50*ln(2)*ln(x)+25*ln(x)^2-25*I*ln(x)*Pi* 
csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I/(exp(x)+x^2-3*x))*csgn(I* 
((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))-25*I*Pi*ln(x^2-3*x+exp 
(-5)*x^2-3*exp(-5)*x+exp(x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csg 
n(I/(exp(x)+x^2-3*x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2 
-3*x))-25*I*ln(x)*Pi*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2- 
3*x))*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I* 
x)+25*I*Pi*ln(exp(x)+x^2-3*x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*cs 
gn(I/(exp(x)+x^2-3*x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^ 
2-3*x))+25*I*Pi*ln(exp(x)+x^2-3*x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x 
)/(exp(x)+x^2-3*x))*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2 
-3*x))*csgn(I*x)+(50*ln(x)-50*ln(exp(x)+x^2-3*x))*ln((exp(x)+x^2-3*x)*exp( 
5)+x^2-3*x)-50*ln(exp(x)+x^2-3*x)*ln(2)-50*ln(x)*ln(exp(x)+x^2-3*x)+25*I*P 
i*ln(x^2-3*x+exp(-5)*x^2-3*exp(-5)*x+exp(x))*csgn(I/(exp(x)+x^2-3*x))*csgn 
(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))^2+25*I*Pi*ln(x^2-3* 
x+exp(-5)*x^2-3*exp(-5)*x+exp(x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x) 
/(exp(x)+x^2-3*x))*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2- 
3*x))^2+25*I*Pi*ln(x^2-3*x+exp(-5)*x^2-3*exp(-5)*x+exp(x))*csgn(I*x/(ex...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \, \log \left (\frac {2 \, {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{10} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 10\right )}\right )}}{{\left (x^{2} - 3 \, x\right )} e^{5} + e^{\left (x + 5\right )}}\right )^{2} \] Input:

integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x 
)*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* 
exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 
)*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 
)+x^5-6*x^4+9*x^3),x, algorithm="fricas")
 

Output:

25*log(2*((x^3 - 3*x^2)*e^10 + (x^3 - 3*x^2)*e^5 + x*e^(x + 10))/((x^2 - 3 
*x)*e^5 + e^(x + 5)))^2
 

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \log {\left (\frac {2 x^{3} - 6 x^{2} + 2 x e^{5} e^{x} + \left (2 x^{3} - 6 x^{2}\right ) e^{5}}{x^{2} - 3 x + e^{x}} \right )}^{2} \] Input:

integrate((50*exp(5)*exp(x)**2+((100*x**2-300*x)*exp(5)-50*x**3+300*x**2-3 
00*x)*exp(x)+(50*x**4-300*x**3+450*x**2)*exp(5)+50*x**4-300*x**3+450*x**2) 
*ln((2*x*exp(5)*exp(x)+(2*x**3-6*x**2)*exp(5)+2*x**3-6*x**2)/(exp(x)+x**2- 
3*x))/(x*exp(5)*exp(x)**2+((2*x**3-6*x**2)*exp(5)+x**3-3*x**2)*exp(x)+(x** 
5-6*x**4+9*x**3)*exp(5)+x**5-6*x**4+9*x**3),x)
 

Output:

25*log((2*x**3 - 6*x**2 + 2*x*exp(5)*exp(x) + (2*x**3 - 6*x**2)*exp(5))/(x 
**2 - 3*x + exp(x)))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (28) = 56\).

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 5.91 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=50 \, {\left (\log \left (x^{2} - 3 \, x + e^{x}\right ) - \log \left (x\right ) + 5\right )} \log \left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right ) - 25 \, \log \left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right )^{2} + 50 \, {\left (\log \left (x\right ) - 5\right )} \log \left (x^{2} - 3 \, x + e^{x}\right ) - 25 \, \log \left (x^{2} - 3 \, x + e^{x}\right )^{2} - 25 \, \log \left (x\right )^{2} - 50 \, {\left (\log \left (x^{2} - 3 \, x + e^{x}\right ) - \log \left ({\left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right )} e^{\left (-5\right )}\right ) - \log \left (x\right )\right )} \log \left (\frac {2 \, {\left (x^{3} - 3 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 5\right )}\right )}}{x^{2} - 3 \, x + e^{x}}\right ) + 250 \, \log \left (x\right ) \] Input:

integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x 
)*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* 
exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 
)*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 
)+x^5-6*x^4+9*x^3),x, algorithm="maxima")
 

Output:

50*(log(x^2 - 3*x + e^x) - log(x) + 5)*log(x^2*(e^5 + 1) - 3*x*(e^5 + 1) + 
 e^(x + 5)) - 25*log(x^2*(e^5 + 1) - 3*x*(e^5 + 1) + e^(x + 5))^2 + 50*(lo 
g(x) - 5)*log(x^2 - 3*x + e^x) - 25*log(x^2 - 3*x + e^x)^2 - 25*log(x)^2 - 
 50*(log(x^2 - 3*x + e^x) - log((x^2*(e^5 + 1) - 3*x*(e^5 + 1) + e^(x + 5) 
)*e^(-5)) - log(x))*log(2*(x^3 - 3*x^2 + (x^3 - 3*x^2)*e^5 + x*e^(x + 5))/ 
(x^2 - 3*x + e^x)) + 250*log(x)
 

Giac [F]

\[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=\int { \frac {50 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2} + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{5} - {\left (x^{3} - 6 \, x^{2} - 2 \, {\left (x^{2} - 3 \, x\right )} e^{5} + 6 \, x\right )} e^{x} + e^{\left (2 \, x + 5\right )}\right )} \log \left (\frac {2 \, {\left (x^{3} - 3 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 5\right )}\right )}}{x^{2} - 3 \, x + e^{x}}\right )}{x^{5} - 6 \, x^{4} + 9 \, x^{3} + {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} e^{5} + x e^{\left (2 \, x + 5\right )} + {\left (x^{3} - 3 \, x^{2} + 2 \, {\left (x^{3} - 3 \, x^{2}\right )} e^{5}\right )} e^{x}} \,d x } \] Input:

integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x 
)*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* 
exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 
)*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 
)+x^5-6*x^4+9*x^3),x, algorithm="giac")
 

Output:

integrate(50*(x^4 - 6*x^3 + 9*x^2 + (x^4 - 6*x^3 + 9*x^2)*e^5 - (x^3 - 6*x 
^2 - 2*(x^2 - 3*x)*e^5 + 6*x)*e^x + e^(2*x + 5))*log(2*(x^3 - 3*x^2 + (x^3 
 - 3*x^2)*e^5 + x*e^(x + 5))/(x^2 - 3*x + e^x))/(x^5 - 6*x^4 + 9*x^3 + (x^ 
5 - 6*x^4 + 9*x^3)*e^5 + x*e^(2*x + 5) + (x^3 - 3*x^2 + 2*(x^3 - 3*x^2)*e^ 
5)*e^x), x)
 

Mupad [B] (verification not implemented)

Time = 3.56 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25\,{\ln \left (-\frac {{\mathrm {e}}^5\,\left (6\,x^2-2\,x^3\right )+6\,x^2-2\,x^3-2\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{{\mathrm {e}}^x-3\,x+x^2}\right )}^2 \] Input:

int((log(-(exp(5)*(6*x^2 - 2*x^3) + 6*x^2 - 2*x^3 - 2*x*exp(5)*exp(x))/(ex 
p(x) - 3*x + x^2))*(50*exp(2*x)*exp(5) - exp(x)*(300*x + exp(5)*(300*x - 1 
00*x^2) - 300*x^2 + 50*x^3) + exp(5)*(450*x^2 - 300*x^3 + 50*x^4) + 450*x^ 
2 - 300*x^3 + 50*x^4))/(exp(5)*(9*x^3 - 6*x^4 + x^5) - exp(x)*(exp(5)*(6*x 
^2 - 2*x^3) + 3*x^2 - x^3) + 9*x^3 - 6*x^4 + x^5 + x*exp(2*x)*exp(5)),x)
 

Output:

25*log(-(exp(5)*(6*x^2 - 2*x^3) + 6*x^2 - 2*x^3 - 2*x*exp(5)*exp(x))/(exp( 
x) - 3*x + x^2))^2
 

Reduce [F]

\[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=\int \frac {\left (50 \,{\mathrm e}^{5} \left ({\mathrm e}^{x}\right )^{2}+\left (\left (100 x^{2}-300 x \right ) {\mathrm e}^{5}-50 x^{3}+300 x^{2}-300 x \right ) {\mathrm e}^{x}+\left (50 x^{4}-300 x^{3}+450 x^{2}\right ) {\mathrm e}^{5}+50 x^{4}-300 x^{3}+450 x^{2}\right ) \mathrm {log}\left (\frac {2 x \,{\mathrm e}^{5} {\mathrm e}^{x}+\left (2 x^{3}-6 x^{2}\right ) {\mathrm e}^{5}+2 x^{3}-6 x^{2}}{{\mathrm e}^{x}+x^{2}-3 x}\right )}{x \,{\mathrm e}^{5} \left ({\mathrm e}^{x}\right )^{2}+\left (\left (2 x^{3}-6 x^{2}\right ) {\mathrm e}^{5}+x^{3}-3 x^{2}\right ) {\mathrm e}^{x}+\left (x^{5}-6 x^{4}+9 x^{3}\right ) {\mathrm e}^{5}+x^{5}-6 x^{4}+9 x^{3}}d x \] Input:

int((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x)*exp( 
x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x*exp(5) 
*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5)*exp( 
x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5)+x^5- 
6*x^4+9*x^3),x)
 

Output:

int((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x)*exp( 
x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x*exp(5) 
*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5)*exp( 
x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5)+x^5- 
6*x^4+9*x^3),x)