\(\int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} (80 x-40 x^2-80 x^3)+(400+400 x^2+160 e^{2 x} x^2) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} (-10 x^3+50 x^4+10 x^5)+(-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} (40 x^2-200 x^3-40 x^4)) \log (x)+(100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} (-40 x+200 x^2+40 x^3)) \log ^2(x)} \, dx\) [765]

Optimal result

Integrand size = 229, antiderivative size = 31 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {4}{\left (-5-\frac {e^{2 x}}{5}+\frac {1}{x}-x\right ) \left (-\frac {x}{2}+\log (x)\right )} \] Output:

4/(1/x-x-5-1/5*exp(2*x))/(ln(x)-1/2*x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=-\frac {40 x}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (-x+2 \log (x))} \] Input:

Integrate[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80 
*x^3) + (400 + 400*x^2 + 160*E^(2*x)*x^2)*Log[x])/(25*x^2 - 250*x^3 + 575* 
x^4 + E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^5) 
 + (-100*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^ 
(2*x)*(40*x^2 - 200*x^3 - 40*x^4))*Log[x] + (100 - 1000*x + 2300*x^2 + 4*E 
^(4*x)*x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*Log[ 
x]^2),x]
 

Output:

(-40*x)/((-5 + 25*x + E^(2*x)*x + 5*x^2)*(-x + 2*Log[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.

\(\displaystyle \int \frac {-400 x^3-600 x^2+\left (160 e^{2 x} x^2+400 x^2+400\right ) \log (x)+e^{2 x} \left (-80 x^3-40 x^2+80 x\right )+2000 x-400}{25 x^6+250 x^5+e^{4 x} x^4+575 x^4-250 x^3+25 x^2+e^{2 x} \left (10 x^5+50 x^4-10 x^3\right )+\left (100 x^4+1000 x^3+4 e^{4 x} x^2+2300 x^2+e^{2 x} \left (40 x^3+200 x^2-40 x\right )-1000 x+100\right ) \log ^2(x)+\left (-100 x^5-1000 x^4-4 e^{4 x} x^3-2300 x^3+1000 x^2+e^{2 x} \left (-40 x^4-200 x^3+40 x^2\right )-100 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {40 \left (-2 \left (e^{2 x}+5\right ) x^3-\left (e^{2 x}+15\right ) x^2+2 \left (\left (2 e^{2 x}+5\right ) x^2+5\right ) \log (x)+2 \left (e^{2 x}+25\right ) x-10\right )}{\left (-5 x^2-\left (e^{2 x}+25\right ) x+5\right )^2 (x-2 \log (x))^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 40 \int -\frac {2 \left (5+e^{2 x}\right ) x^3+\left (15+e^{2 x}\right ) x^2-2 \left (25+e^{2 x}\right ) x-2 \left (\left (5+2 e^{2 x}\right ) x^2+5\right ) \log (x)+10}{\left (-5 x^2-\left (25+e^{2 x}\right ) x+5\right )^2 (x-2 \log (x))^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -40 \int \frac {2 \left (5+e^{2 x}\right ) x^3+\left (15+e^{2 x}\right ) x^2-2 \left (25+e^{2 x}\right ) x-2 \left (\left (5+2 e^{2 x}\right ) x^2+5\right ) \log (x)+10}{\left (-5 x^2-\left (25+e^{2 x}\right ) x+5\right )^2 (x-2 \log (x))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -40 \int \left (\frac {2 x^2-4 \log (x) x+x-2}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}-\frac {5 \left (2 x^3+9 x^2-2 x-1\right )}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -40 \left (-2 \int \frac {1}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+\int \frac {x}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+2 \int \frac {x^2}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+5 \int \frac {1}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx+10 \int \frac {x}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx-45 \int \frac {x^2}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx-4 \int \frac {x \log (x)}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx-10 \int \frac {x^3}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx\right )\)

Input:

Int[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80*x^3) 
+ (400 + 400*x^2 + 160*E^(2*x)*x^2)*Log[x])/(25*x^2 - 250*x^3 + 575*x^4 + 
E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^5) + (-1 
00*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^(2*x)* 
(40*x^2 - 200*x^3 - 40*x^4))*Log[x] + (100 - 1000*x + 2300*x^2 + 4*E^(4*x) 
*x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*Log[x]^2), 
x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {40 x}{\left (5 x^{2}+x \,{\mathrm e}^{2 x}+25 x -5\right ) \left (x -2 \ln \left (x \right )\right )}\)	\(30\)
parallelrisch	\(\frac {40 x}{5 x^{3}+{\mathrm e}^{2 x} x^{2}-10 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{2 x} \ln \left (x \right ) x +25 x^{2}-50 x \ln \left (x \right )-5 x +10 \ln \left (x \right )}\)	\(53\)

Input:

int(((160*exp(2*x)*x^2+400*x^2+400)*ln(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-4 
00*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2* 
x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*ln(x)^2+(-4*x^3*exp(2*x)^2+(-40*x 
^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*ln(x 
)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-25 
0*x^3+25*x^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 25 \, x^{2} - 2 \, {\left (5 \, x^{2} + x e^{\left (2 \, x\right )} + 25 \, x - 5\right )} \log \left (x\right ) - 5 \, x} \] Input:

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp 
(2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) 
*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 
2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 
*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 
75*x^4-250*x^3+25*x^2),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 x}{5 x^{3} - 10 x^{2} \log {\left (x \right )} + 25 x^{2} - 50 x \log {\left (x \right )} - 5 x + \left (x^{2} - 2 x \log {\left (x \right )}\right ) e^{2 x} + 10 \log {\left (x \right )}} \] Input:

integrate(((160*exp(2*x)*x**2+400*x**2+400)*ln(x)+(-80*x**3-40*x**2+80*x)* 
exp(2*x)-400*x**3-600*x**2+2000*x-400)/((4*x**2*exp(2*x)**2+(40*x**3+200*x 
**2-40*x)*exp(2*x)+100*x**4+1000*x**3+2300*x**2-1000*x+100)*ln(x)**2+(-4*x 
**3*exp(2*x)**2+(-40*x**4-200*x**3+40*x**2)*exp(2*x)-100*x**5-1000*x**4-23 
00*x**3+1000*x**2-100*x)*ln(x)+x**4*exp(2*x)**2+(10*x**5+50*x**4-10*x**3)* 
exp(2*x)+25*x**6+250*x**5+575*x**4-250*x**3+25*x**2),x)
 

Output:

40*x/(5*x**3 - 10*x**2*log(x) + 25*x**2 - 50*x*log(x) - 5*x + (x**2 - 2*x* 
log(x))*exp(2*x) + 10*log(x))
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 2 \, x \log \left (x\right )\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{2} + 5 \, x - 1\right )} \log \left (x\right ) - 5 \, x} \] Input:

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp 
(2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) 
*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 
2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 
*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 
75*x^4-250*x^3+25*x^2),x, algorithm="maxima")
 

Output:

40*x/(5*x^3 + 25*x^2 + (x^2 - 2*x*log(x))*e^(2*x) - 10*(x^2 + 5*x - 1)*log 
(x) - 5*x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 10 \, x^{2} \log \left (x\right ) - 2 \, x e^{\left (2 \, x\right )} \log \left (x\right ) + 25 \, x^{2} - 50 \, x \log \left (x\right ) - 5 \, x + 10 \, \log \left (x\right )} \] Input:

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp 
(2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) 
*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 
2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 
*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 
75*x^4-250*x^3+25*x^2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=-\int \frac {{\mathrm {e}}^{2\,x}\,\left (80\,x^3+40\,x^2-80\,x\right )-2000\,x-\ln \left (x\right )\,\left (160\,x^2\,{\mathrm {e}}^{2\,x}+400\,x^2+400\right )+600\,x^2+400\,x^3+400}{{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (40\,x^3+200\,x^2-40\,x\right )-1000\,x+4\,x^2\,{\mathrm {e}}^{4\,x}+2300\,x^2+1000\,x^3+100\,x^4+100\right )-\ln \left (x\right )\,\left (100\,x+4\,x^3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (40\,x^4+200\,x^3-40\,x^2\right )-1000\,x^2+2300\,x^3+1000\,x^4+100\,x^5\right )+x^4\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (10\,x^5+50\,x^4-10\,x^3\right )+25\,x^2-250\,x^3+575\,x^4+250\,x^5+25\,x^6} \,d x \] Input:

int(-(exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x 
) + 400*x^2 + 400) + 600*x^2 + 400*x^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 
 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*x^ 
4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 4 
0*x^4) - 1000*x^2 + 2300*x^3 + 1000*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2* 
x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 + 25* 
x^6),x)
 

Output:

-int((exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x 
) + 400*x^2 + 400) + 600*x^2 + 400*x^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 
 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*x^ 
4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 4 
0*x^4) - 1000*x^2 + 2300*x^3 + 1000*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2* 
x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 + 25* 
x^6), x)
 

Reduce [F]

\[ \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\text {too large to display} \] Input:

int(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)- 
400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2 
*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^2+(-40 
*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*lo 
g(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4 
-250*x^3+25*x^2),x)
 

Output:

40*( - 10*int(x**3/(4*e**(4*x)*log(x)**2*x**2 - 4*e**(4*x)*log(x)*x**3 + e 
**(4*x)*x**4 + 40*e**(2*x)*log(x)**2*x**3 + 200*e**(2*x)*log(x)**2*x**2 - 
40*e**(2*x)*log(x)**2*x - 40*e**(2*x)*log(x)*x**4 - 200*e**(2*x)*log(x)*x* 
*3 + 40*e**(2*x)*log(x)*x**2 + 10*e**(2*x)*x**5 + 50*e**(2*x)*x**4 - 10*e* 
*(2*x)*x**3 + 100*log(x)**2*x**4 + 1000*log(x)**2*x**3 + 2300*log(x)**2*x* 
*2 - 1000*log(x)**2*x + 100*log(x)**2 - 100*log(x)*x**5 - 1000*log(x)*x**4 
 - 2300*log(x)*x**3 + 1000*log(x)*x**2 - 100*log(x)*x + 25*x**6 + 250*x**5 
 + 575*x**4 - 250*x**3 + 25*x**2),x) - 15*int(x**2/(4*e**(4*x)*log(x)**2*x 
**2 - 4*e**(4*x)*log(x)*x**3 + e**(4*x)*x**4 + 40*e**(2*x)*log(x)**2*x**3 
+ 200*e**(2*x)*log(x)**2*x**2 - 40*e**(2*x)*log(x)**2*x - 40*e**(2*x)*log( 
x)*x**4 - 200*e**(2*x)*log(x)*x**3 + 40*e**(2*x)*log(x)*x**2 + 10*e**(2*x) 
*x**5 + 50*e**(2*x)*x**4 - 10*e**(2*x)*x**3 + 100*log(x)**2*x**4 + 1000*lo 
g(x)**2*x**3 + 2300*log(x)**2*x**2 - 1000*log(x)**2*x + 100*log(x)**2 - 10 
0*log(x)*x**5 - 1000*log(x)*x**4 - 2300*log(x)*x**3 + 1000*log(x)*x**2 - 1 
00*log(x)*x + 25*x**6 + 250*x**5 + 575*x**4 - 250*x**3 + 25*x**2),x) + 10* 
int(log(x)/(4*e**(4*x)*log(x)**2*x**2 - 4*e**(4*x)*log(x)*x**3 + e**(4*x)* 
x**4 + 40*e**(2*x)*log(x)**2*x**3 + 200*e**(2*x)*log(x)**2*x**2 - 40*e**(2 
*x)*log(x)**2*x - 40*e**(2*x)*log(x)*x**4 - 200*e**(2*x)*log(x)*x**3 + 40* 
e**(2*x)*log(x)*x**2 + 10*e**(2*x)*x**5 + 50*e**(2*x)*x**4 - 10*e**(2*x)*x 
**3 + 100*log(x)**2*x**4 + 1000*log(x)**2*x**3 + 2300*log(x)**2*x**2 - ...