\(\int \frac {2500+1000 x+100 x^2+e^x (500-300 x-180 x^2-20 x^3)+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x (390625 x+156250 x^2+15625 x^3)+(-15625 x-3125 x^2) \log (3)+(1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x (390625 x+156250 x^2+15625 x^3)+(-15625 x-3125 x^2) \log (3)) \log (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(781250 x+343750 x^2+43750 x^3+1250 x^4+e^x (156250 x+62500 x^2+6250 x^3)+(-6250 x-1250 x^2) \log (3)) \log ^2(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(156250 x+68750 x^2+8750 x^3+250 x^4+e^x (31250 x+12500 x^2+1250 x^3)+(-1250 x-250 x^2) \log (3)) \log ^3(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(15625 x+6875 x^2+875 x^3+25 x^4+e^x (3125 x+1250 x^2+125 x^3)+(-125 x-25 x^2) \log (3)) \log ^4(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(625 x+275 x^2+35 x^3+x^4+e^x (125 x+50 x^2+5 x^3)+(-5 x-x^2) \log (3)) \log ^5(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})} \, dx\) [2837]

Optimal result

Integrand size = 505, antiderivative size = 27 \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\frac {1}{\left (5+\log \left (\frac {5 \left (5+e^x\right )+x-\frac {\log (3)}{5+x}}{x}\right )\right )^4} \] Output:

1/(ln((x-ln(3)/(5+x)+5*exp(x)+25)/x)+5)^4
 

Mathematica [F]

\[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx \] Input:

Integrate[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) 
+ (-20 - 8*x)*Log[3])/(1953125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^ 
x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (19 
53125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 
+ 15625*x^3) + (-15625*x - 3125*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*( 
25 + 5*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3 + 12 
50*x^4 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[ 
3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^2 + (156 
250*x + 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 + 1250*x 
^3) + (-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - 
 Log[3])/(5*x + x^2)]^3 + (15625*x + 6875*x^2 + 875*x^3 + 25*x^4 + E^x*(31 
25*x + 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 + 30*x + x 
^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + 
 x^4 + E^x*(125*x + 50*x^2 + 5*x^3) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x 
 + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5),x]
 

Output:

Integrate[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) 
+ (-20 - 8*x)*Log[3])/(1953125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^ 
x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (19 
53125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 
+ 15625*x^3) + (-15625*x - 3125*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*( 
25 + 5*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3 + 12 
50*x^4 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[ 
3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^2 + (156 
250*x + 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 + 1250*x 
^3) + (-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - 
 Log[3])/(5*x + x^2)]^3 + (15625*x + 6875*x^2 + 875*x^3 + 25*x^4 + E^x*(31 
25*x + 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 + 30*x + x 
^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + 
 x^4 + E^x*(125*x + 50*x^2 + 5*x^3) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x 
 + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5), 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.

\(\displaystyle \int \frac {100 x^2+e^x \left (-20 x^3-180 x^2-300 x+500\right )+1000 x+(-8 x-20) \log (3)+2500}{3125 x^4+109375 x^3+859375 x^2+\left (-3125 x^2-15625 x\right ) \log (3)+e^x \left (15625 x^3+156250 x^2+390625 x\right )+\left (x^4+35 x^3+275 x^2+\left (-x^2-5 x\right ) \log (3)+e^x \left (5 x^3+50 x^2+125 x\right )+625 x\right ) \log ^5\left (\frac {x^2+30 x+e^x (5 x+25)+125-\log (3)}{x^2+5 x}\right )+\left (25 x^4+875 x^3+6875 x^2+\left (-25 x^2-125 x\right ) \log (3)+e^x \left (125 x^3+1250 x^2+3125 x\right )+15625 x\right ) \log ^4\left (\frac {x^2+30 x+e^x (5 x+25)+125-\log (3)}{x^2+5 x}\right )+\left (250 x^4+8750 x^3+68750 x^2+\left (-250 x^2-1250 x\right ) \log (3)+e^x \left (1250 x^3+12500 x^2+31250 x\right )+156250 x\right ) \log ^3\left (\frac {x^2+30 x+e^x (5 x+25)+125-\log (3)}{x^2+5 x}\right )+\left (1250 x^4+43750 x^3+343750 x^2+\left (-1250 x^2-6250 x\right ) \log (3)+e^x \left (6250 x^3+62500 x^2+156250 x\right )+781250 x\right ) \log ^2\left (\frac {x^2+30 x+e^x (5 x+25)+125-\log (3)}{x^2+5 x}\right )+\left (3125 x^4+109375 x^3+859375 x^2+\left (-3125 x^2-15625 x\right ) \log (3)+e^x \left (15625 x^3+156250 x^2+390625 x\right )+1953125 x\right ) \log \left (\frac {x^2+30 x+e^x (5 x+25)+125-\log (3)}{x^2+5 x}\right )+1953125 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (25 x^2-5 e^x (x-1) (x+5)^2-x (\log (9)-250)+625 \left (1-\frac {\log (3)}{125}\right )\right )}{x (x+5) \left (x^2+30 x+5 e^x (x+5)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)+125-\log (3)}{x (x+5)}\right )+5\right )^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int \frac {25 x^2+(250-\log (9)) x+5 e^x (1-x) (x+5)^2+5 (125-\log (3))}{x (x+5) \left (x^2+30 x+5 e^x (x+5)-\log (3)+125\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle 4 \int \frac {25 x^2+(250-\log (9)) x+5 e^x (1-x) (x+5)^2+5 (125-\log (3))}{x (x+5) \left (x^2+30 x+5 e^x (x+5)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 \int \left (\frac {1-x}{x \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}+\frac {x^3+34 x^2+(265-\log (3)) x-\log (729)+600}{(x+5) \left (x^2+5 e^x x+30 x+25 e^x+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \left (-\int \frac {1}{\left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx+\int \frac {1}{x \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx+(120-\log (3)) \int \frac {1}{\left (x^2+5 e^x x+30 x+25 e^x+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx+\log (3) \int \frac {1}{(-x-5) \left (x^2+5 e^x x+30 x+25 e^x+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx+29 \int \frac {x}{\left (x^2+5 e^x x+30 x+25 e^x+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx+\int \frac {x^2}{\left (x^2+5 e^x x+30 x+25 e^x+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (\log \left (\frac {x^2+30 x+5 e^x (x+5)-\log (3)+125}{x (x+5)}\right )+5\right )^5}dx\right )\)

Input:

Int[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) + (-20 
 - 8*x)*Log[3])/(1953125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390 
625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (1953125* 
x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 1562 
5*x^3) + (-15625*x - 3125*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5 
*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3 + 1250*x^4 
 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[3])*Lo 
g[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^2 + (156250*x 
+ 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 + 1250*x^3) + 
(-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3 
])/(5*x + x^2)]^3 + (15625*x + 6875*x^2 + 875*x^3 + 25*x^4 + E^x*(3125*x + 
 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E 
^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + x^4 + 
 E^x*(125*x + 50*x^2 + 5*x^3) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x + x^2 
 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (-20 x^{3}-180 x^{2}-300 x +500\right ) {\mathrm e}^{x}+\left (-8 x -20\right ) \ln \left (3\right )+100 x^{2}+1000 x +2500}{\left (\left (5 x^{3}+50 x^{2}+125 x \right ) {\mathrm e}^{x}+\left (-x^{2}-5 x \right ) \ln \left (3\right )+x^{4}+35 x^{3}+275 x^{2}+625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{5}+\left (\left (125 x^{3}+1250 x^{2}+3125 x \right ) {\mathrm e}^{x}+\left (-25 x^{2}-125 x \right ) \ln \left (3\right )+25 x^{4}+875 x^{3}+6875 x^{2}+15625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{4}+\left (\left (1250 x^{3}+12500 x^{2}+31250 x \right ) {\mathrm e}^{x}+\left (-250 x^{2}-1250 x \right ) \ln \left (3\right )+250 x^{4}+8750 x^{3}+68750 x^{2}+156250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{3}+\left (\left (6250 x^{3}+62500 x^{2}+156250 x \right ) {\mathrm e}^{x}+\left (-1250 x^{2}-6250 x \right ) \ln \left (3\right )+1250 x^{4}+43750 x^{3}+343750 x^{2}+781250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{2}+\left (\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \left (3\right )+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )+\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \left (3\right )+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x}d x\]

Input:

int(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x^2+1000*x+250 
0)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*ln(3)+x^4+35*x^3+275*x^2+625*x 
)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3+1250*x^2+ 
3125*x)*exp(x)+(-25*x^2-125*x)*ln(3)+25*x^4+875*x^3+6875*x^2+15625*x)*ln(( 
(25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x^2+3125 
0*x)*exp(x)+(-250*x^2-1250*x)*ln(3)+250*x^4+8750*x^3+68750*x^2+156250*x)*l 
n(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+1 
56250*x)*exp(x)+(-1250*x^2-6250*x)*ln(3)+1250*x^4+43750*x^3+343750*x^2+781 
250*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^2+((15625*x^3+15 
6250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109375*x^3+85 
9375*x^2+1953125*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))+(15 
625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109 
375*x^3+859375*x^2+1953125*x),x)
 

Output:

int(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x^2+1000*x+250 
0)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*ln(3)+x^4+35*x^3+275*x^2+625*x 
)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3+1250*x^2+ 
3125*x)*exp(x)+(-25*x^2-125*x)*ln(3)+25*x^4+875*x^3+6875*x^2+15625*x)*ln(( 
(25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x^2+3125 
0*x)*exp(x)+(-250*x^2-1250*x)*ln(3)+250*x^4+8750*x^3+68750*x^2+156250*x)*l 
n(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+1 
56250*x)*exp(x)+(-1250*x^2-6250*x)*ln(3)+1250*x^4+43750*x^3+343750*x^2+781 
250*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^2+((15625*x^3+15 
6250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109375*x^3+85 
9375*x^2+1953125*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))+(15 
625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109 
375*x^3+859375*x^2+1953125*x),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (25) = 50\).

Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.04 \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\frac {1}{\log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125}{x^{2} + 5 \, x}\right )^{4} + 20 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125}{x^{2} + 5 \, x}\right )^{3} + 150 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125}{x^{2} + 5 \, x}\right )^{2} + 500 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125}{x^{2} + 5 \, x}\right ) + 625} \] Input:

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+100 
0*x+2500)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x 
^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3 
+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15 
625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+1 
2500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+68750*x 
^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250 
*x^3+62500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x 
^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x 
))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+31 
25*x^4+109375*x^3+859375*x^2+1953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30 
*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625 
*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="fricas" 
)
 

Output:

1/(log((x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x))^4 + 20*log 
((x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x))^3 + 150*log((x^2 
 + 5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x))^2 + 500*log((x^2 + 5* 
(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x)) + 625)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (22) = 44\).

Time = 0.70 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.74 \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\frac {1}{\log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\left (3 \right )} + 125}{x^{2} + 5 x} \right )}^{4} + 20 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\left (3 \right )} + 125}{x^{2} + 5 x} \right )}^{3} + 150 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\left (3 \right )} + 125}{x^{2} + 5 x} \right )}^{2} + 500 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\left (3 \right )} + 125}{x^{2} + 5 x} \right )} + 625} \] Input:

integrate(((-20*x**3-180*x**2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x**2+1 
000*x+2500)/(((5*x**3+50*x**2+125*x)*exp(x)+(-x**2-5*x)*ln(3)+x**4+35*x**3 
+275*x**2+625*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))**5+( 
(125*x**3+1250*x**2+3125*x)*exp(x)+(-25*x**2-125*x)*ln(3)+25*x**4+875*x**3 
+6875*x**2+15625*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))** 
4+((1250*x**3+12500*x**2+31250*x)*exp(x)+(-250*x**2-1250*x)*ln(3)+250*x**4 
+8750*x**3+68750*x**2+156250*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/( 
x**2+5*x))**3+((6250*x**3+62500*x**2+156250*x)*exp(x)+(-1250*x**2-6250*x)* 
ln(3)+1250*x**4+43750*x**3+343750*x**2+781250*x)*ln(((25+5*x)*exp(x)-ln(3) 
+x**2+30*x+125)/(x**2+5*x))**2+((15625*x**3+156250*x**2+390625*x)*exp(x)+( 
-3125*x**2-15625*x)*ln(3)+3125*x**4+109375*x**3+859375*x**2+1953125*x)*ln( 
((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))+(15625*x**3+156250*x**2+ 
390625*x)*exp(x)+(-3125*x**2-15625*x)*ln(3)+3125*x**4+109375*x**3+859375*x 
**2+1953125*x),x)
 

Output:

1/(log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x))**4 + 
 20*log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x))**3 
+ 150*log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x))** 
2 + 500*log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x)) 
 + 625)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (25) = 50\).

Time = 19.21 (sec) , antiderivative size = 266, normalized size of antiderivative = 9.85 \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=-\frac {1}{4 \, {\left (\log \left (x + 5\right ) + \log \left (x\right ) - 5\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125\right )^{3} - \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125\right )^{4} - 4 \, {\left (\log \left (x\right ) - 5\right )} \log \left (x + 5\right )^{3} - \log \left (x + 5\right )^{4} - \log \left (x\right )^{4} - 6 \, {\left (2 \, {\left (\log \left (x\right ) - 5\right )} \log \left (x + 5\right ) + \log \left (x + 5\right )^{2} + \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 25\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125\right )^{2} - 6 \, {\left (\log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 25\right )} \log \left (x + 5\right )^{2} + 20 \, \log \left (x\right )^{3} + 4 \, {\left (3 \, {\left (\log \left (x\right ) - 5\right )} \log \left (x + 5\right )^{2} + \log \left (x + 5\right )^{3} + \log \left (x\right )^{3} + 3 \, {\left (\log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 25\right )} \log \left (x + 5\right ) - 15 \, \log \left (x\right )^{2} + 75 \, \log \left (x\right ) - 125\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \left (3\right ) + 125\right ) - 4 \, {\left (\log \left (x\right )^{3} - 15 \, \log \left (x\right )^{2} + 75 \, \log \left (x\right ) - 125\right )} \log \left (x + 5\right ) - 150 \, \log \left (x\right )^{2} + 500 \, \log \left (x\right ) - 625} \] Input:

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+100 
0*x+2500)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x 
^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3 
+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15 
625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+1 
2500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+68750*x 
^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250 
*x^3+62500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x 
^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x 
))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+31 
25*x^4+109375*x^3+859375*x^2+1953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30 
*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625 
*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="maxima" 
)
 

Output:

-1/(4*(log(x + 5) + log(x) - 5)*log(x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 
125)^3 - log(x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)^4 - 4*(log(x) - 5) 
*log(x + 5)^3 - log(x + 5)^4 - log(x)^4 - 6*(2*(log(x) - 5)*log(x + 5) + l 
og(x + 5)^2 + log(x)^2 - 10*log(x) + 25)*log(x^2 + 5*(x + 5)*e^x + 30*x - 
log(3) + 125)^2 - 6*(log(x)^2 - 10*log(x) + 25)*log(x + 5)^2 + 20*log(x)^3 
 + 4*(3*(log(x) - 5)*log(x + 5)^2 + log(x + 5)^3 + log(x)^3 + 3*(log(x)^2 
- 10*log(x) + 25)*log(x + 5) - 15*log(x)^2 + 75*log(x) - 125)*log(x^2 + 5* 
(x + 5)*e^x + 30*x - log(3) + 125) - 4*(log(x)^3 - 15*log(x)^2 + 75*log(x) 
 - 125)*log(x + 5) - 150*log(x)^2 + 500*log(x) - 625)
 

Giac [F(-1)]

Timed out. \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\text {Timed out} \] Input:

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+100 
0*x+2500)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x 
^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3 
+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15 
625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+1 
2500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+68750*x 
^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250 
*x^3+62500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x 
^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x 
))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+31 
25*x^4+109375*x^3+859375*x^2+1953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30 
*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625 
*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\int \frac {1000\,x-\ln \left (3\right )\,\left (8\,x+20\right )+100\,x^2-{\mathrm {e}}^x\,\left (20\,x^3+180\,x^2+300\,x-500\right )+2500}{1953125\,x-\ln \left (3\right )\,\left (3125\,x^2+15625\,x\right )+{\ln \left (\frac {30\,x-\ln \left (3\right )+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^4\,\left (15625\,x-\ln \left (3\right )\,\left (25\,x^2+125\,x\right )+6875\,x^2+875\,x^3+25\,x^4+{\mathrm {e}}^x\,\left (125\,x^3+1250\,x^2+3125\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \left (3\right )+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^3\,\left (156250\,x-\ln \left (3\right )\,\left (250\,x^2+1250\,x\right )+68750\,x^2+8750\,x^3+250\,x^4+{\mathrm {e}}^x\,\left (1250\,x^3+12500\,x^2+31250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \left (3\right )+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^2\,\left (781250\,x-\ln \left (3\right )\,\left (1250\,x^2+6250\,x\right )+343750\,x^2+43750\,x^3+1250\,x^4+{\mathrm {e}}^x\,\left (6250\,x^3+62500\,x^2+156250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \left (3\right )+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^5\,\left (625\,x+275\,x^2+35\,x^3+x^4-\ln \left (3\right )\,\left (x^2+5\,x\right )+{\mathrm {e}}^x\,\left (5\,x^3+50\,x^2+125\,x\right )\right )+859375\,x^2+109375\,x^3+3125\,x^4+\ln \left (\frac {30\,x-\ln \left (3\right )+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )\,\left (1953125\,x-\ln \left (3\right )\,\left (3125\,x^2+15625\,x\right )+859375\,x^2+109375\,x^3+3125\,x^4+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )\right )+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )} \,d x \] Input:

int((1000*x - log(3)*(8*x + 20) + 100*x^2 - exp(x)*(300*x + 180*x^2 + 20*x 
^3 - 500) + 2500)/(1953125*x - log(3)*(15625*x + 3125*x^2) + log((30*x - l 
og(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^4*(15625*x - log(3)*(1 
25*x + 25*x^2) + 6875*x^2 + 875*x^3 + 25*x^4 + exp(x)*(3125*x + 1250*x^2 + 
 125*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^ 
2))^3*(156250*x - log(3)*(1250*x + 250*x^2) + 68750*x^2 + 8750*x^3 + 250*x 
^4 + exp(x)*(31250*x + 12500*x^2 + 1250*x^3)) + log((30*x - log(3) + exp(x 
)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^2*(781250*x - log(3)*(6250*x + 1250 
*x^2) + 343750*x^2 + 43750*x^3 + 1250*x^4 + exp(x)*(156250*x + 62500*x^2 + 
 6250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x 
^2))^5*(625*x + 275*x^2 + 35*x^3 + x^4 - log(3)*(5*x + x^2) + exp(x)*(125* 
x + 50*x^2 + 5*x^3)) + 859375*x^2 + 109375*x^3 + 3125*x^4 + log((30*x - lo 
g(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))*(1953125*x - log(3)*(15 
625*x + 3125*x^2) + 859375*x^2 + 109375*x^3 + 3125*x^4 + exp(x)*(390625*x 
+ 156250*x^2 + 15625*x^3)) + exp(x)*(390625*x + 156250*x^2 + 15625*x^3)),x 
)
 

Output:

int((1000*x - log(3)*(8*x + 20) + 100*x^2 - exp(x)*(300*x + 180*x^2 + 20*x 
^3 - 500) + 2500)/(1953125*x - log(3)*(15625*x + 3125*x^2) + log((30*x - l 
og(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^4*(15625*x - log(3)*(1 
25*x + 25*x^2) + 6875*x^2 + 875*x^3 + 25*x^4 + exp(x)*(3125*x + 1250*x^2 + 
 125*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^ 
2))^3*(156250*x - log(3)*(1250*x + 250*x^2) + 68750*x^2 + 8750*x^3 + 250*x 
^4 + exp(x)*(31250*x + 12500*x^2 + 1250*x^3)) + log((30*x - log(3) + exp(x 
)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^2*(781250*x - log(3)*(6250*x + 1250 
*x^2) + 343750*x^2 + 43750*x^3 + 1250*x^4 + exp(x)*(156250*x + 62500*x^2 + 
 6250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x 
^2))^5*(625*x + 275*x^2 + 35*x^3 + x^4 - log(3)*(5*x + x^2) + exp(x)*(125* 
x + 50*x^2 + 5*x^3)) + 859375*x^2 + 109375*x^3 + 3125*x^4 + log((30*x - lo 
g(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))*(1953125*x - log(3)*(15 
625*x + 3125*x^2) + 859375*x^2 + 109375*x^3 + 3125*x^4 + exp(x)*(390625*x 
+ 156250*x^2 + 15625*x^3)) + exp(x)*(390625*x + 156250*x^2 + 15625*x^3)), 
x)
 

Reduce [B] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.63 \[ \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx=\frac {1}{\mathrm {log}\left (\frac {5 e^{x} x +25 e^{x}-\mathrm {log}\left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{4}+20 \mathrm {log}\left (\frac {5 e^{x} x +25 e^{x}-\mathrm {log}\left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{3}+150 \mathrm {log}\left (\frac {5 e^{x} x +25 e^{x}-\mathrm {log}\left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )^{2}+500 \,\mathrm {log}\left (\frac {5 e^{x} x +25 e^{x}-\mathrm {log}\left (3\right )+x^{2}+30 x +125}{x^{2}+5 x}\right )+625} \] Input:

int(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+1000*x+25 
00)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x^2+625 
*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3+1250* 
x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15625*x) 
*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x 
^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+68750*x^2+156 
250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+6 
2500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x^3+343 
750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^2+( 
(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+3125*x^4 
+109375*x^3+859375*x^2+1953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125 
)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*lo 
g(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x)
 

Output:

1/(log((5*e**x*x + 25*e**x - log(3) + x**2 + 30*x + 125)/(x**2 + 5*x))**4 
+ 20*log((5*e**x*x + 25*e**x - log(3) + x**2 + 30*x + 125)/(x**2 + 5*x))** 
3 + 150*log((5*e**x*x + 25*e**x - log(3) + x**2 + 30*x + 125)/(x**2 + 5*x) 
)**2 + 500*log((5*e**x*x + 25*e**x - log(3) + x**2 + 30*x + 125)/(x**2 + 5 
*x)) + 625)