\(\int \frac {-15 x^3+10 x^4-x^5+e^x (-250+100 x-260 x^2+100 x^3-10 x^4)+e^{2 x} (50 x^3-20 x^4+2 x^5)+(-15 x+10 x^2-x^3+e^x (250-100 x+10 x^2)+e^{2 x} (50 x-20 x^2+2 x^3)) \log (x)+(1250-500 x+1300 x^2-500 x^3+50 x^4+e^x (-250 x^3+100 x^4-10 x^5)+(-1250+500 x-50 x^2+e^x (-250 x+100 x^2-10 x^3)) \log (x)) \log (\frac {x^2+\log (x)}{2 x})}{25 x^3-10 x^4+x^5+(25 x-10 x^2+x^3) \log (x)} \, dx\) [43]

Optimal result

Integrand size = 231, antiderivative size = 38 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=-2-x+\frac {2 x}{5-x}+\left (-e^x+5 \log \left (\frac {1}{2} \left (x+\frac {\log (x)}{x}\right )\right )\right )^2 \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=e^{2 x}-\frac {10}{-5+x}-x-10 e^x \log \left (\frac {x^2+\log (x)}{2 x}\right )+25 \log ^2\left (\frac {x^2+\log (x)}{2 x}\right ) \] Input:

Integrate[(-15*x^3 + 10*x^4 - x^5 + E^x*(-250 + 100*x - 260*x^2 + 100*x^3 
- 10*x^4) + E^(2*x)*(50*x^3 - 20*x^4 + 2*x^5) + (-15*x + 10*x^2 - x^3 + E^ 
x*(250 - 100*x + 10*x^2) + E^(2*x)*(50*x - 20*x^2 + 2*x^3))*Log[x] + (1250 
 - 500*x + 1300*x^2 - 500*x^3 + 50*x^4 + E^x*(-250*x^3 + 100*x^4 - 10*x^5) 
 + (-1250 + 500*x - 50*x^2 + E^x*(-250*x + 100*x^2 - 10*x^3))*Log[x])*Log[ 
(x^2 + Log[x])/(2*x)])/(25*x^3 - 10*x^4 + x^5 + (25*x - 10*x^2 + x^3)*Log[ 
x]),x]
 

Output:

E^(2*x) - 10/(-5 + x) - x - 10*E^x*Log[(x^2 + Log[x])/(2*x)] + 25*Log[(x^2 
 + Log[x])/(2*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.

\(\displaystyle \int \frac {-x^5+10 x^4-15 x^3+\left (-x^3+10 x^2+e^x \left (10 x^2-100 x+250\right )+e^{2 x} \left (2 x^3-20 x^2+50 x\right )-15 x\right ) \log (x)+e^{2 x} \left (2 x^5-20 x^4+50 x^3\right )+e^x \left (-10 x^4+100 x^3-260 x^2+100 x-250\right )+\left (50 x^4-500 x^3+1300 x^2+\left (-50 x^2+e^x \left (-10 x^3+100 x^2-250 x\right )+500 x-1250\right ) \log (x)+e^x \left (-10 x^5+100 x^4-250 x^3\right )-500 x+1250\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x^5-10 x^4+25 x^3+\left (x^3-10 x^2+25 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-x^5+10 x^4-15 x^3+\left (-x^3+10 x^2+e^x \left (10 x^2-100 x+250\right )+e^{2 x} \left (2 x^3-20 x^2+50 x\right )-15 x\right ) \log (x)+e^{2 x} \left (2 x^5-20 x^4+50 x^3\right )+e^x \left (-10 x^4+100 x^3-260 x^2+100 x-250\right )+\left (50 x^4-500 x^3+1300 x^2+\left (-50 x^2+e^x \left (-10 x^3+100 x^2-250 x\right )+500 x-1250\right ) \log (x)+e^x \left (-10 x^5+100 x^4-250 x^3\right )-500 x+1250\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(5-x)^2 x \left (x^2+\log (x)\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {500 x^2 \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {x^2 \log (x)}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {15 x^2}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {50 x \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {1300 x \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {10 x \log (x)}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {500 \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {500 \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {15 \log (x)}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {1250 \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 x \left (x^2+\log (x)\right )}+\frac {1250 \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 x \left (x^2+\log (x)\right )}-\frac {x^4}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {50 x^3 \log \left (\frac {x^2+\log (x)}{2 x}\right )}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {10 x^3}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {10 e^x \left (x^2+x \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )+x^3 \log \left (\frac {x^2+\log (x)}{2 x}\right )-\log (x)+1\right )}{x \left (x^2+\log (x)\right )}+2 e^{2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 e^{2 x} x^5-x^5-10 e^x x^4-20 e^{2 x} x^4+10 x^4+100 e^x x^3+50 e^{2 x} x^3-15 x^3-260 e^x x^2+\log (x) \left (-x \left (x^2-10 x+15\right )-10 \left (e^x x+5\right ) (x-5)^2 \log \left (\frac {x^2+\log (x)}{2 x}\right )+10 e^x (x-5)^2+2 e^{2 x} x (x-5)^2\right )-10 (x-5)^2 \left (e^x x^3-5 x^2-5\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )+100 e^x x-250 e^x}{(5-x)^2 x \left (x^2+\log (x)\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {15 x^2}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {50 x \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x^2+\log (x)}-\frac {\left (x^2-10 x+15\right ) \log (x)}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {50 \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x \left (x^2+\log (x)\right )}+\frac {50 \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x \left (x^2+\log (x)\right )}-\frac {x^4}{(x-5)^2 \left (x^2+\log (x)\right )}+\frac {10 x^3}{(x-5)^2 \left (x^2+\log (x)\right )}-\frac {10 e^x \left (x^2+x \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )+x^3 \log \left (\frac {x^2+\log (x)}{2 x}\right )-\log (x)+1\right )}{x \left (x^2+\log (x)\right )}+2 e^{2 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 50 \int \frac {\log \left (\frac {x^2+\log (x)}{2 x}\right )}{x \left (x^2+\log (x)\right )}dx+50 \int \frac {x \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x^2+\log (x)}dx-50 \int \frac {\log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{x \left (x^2+\log (x)\right )}dx-\frac {10 e^x \left (x \log (x) \log \left (\frac {x^2+\log (x)}{2 x}\right )+x^3 \log \left (\frac {x^2+\log (x)}{2 x}\right )\right )}{x \left (x^2+\log (x)\right )}-x+e^{2 x}+\frac {10}{5-x}\)

Input:

Int[(-15*x^3 + 10*x^4 - x^5 + E^x*(-250 + 100*x - 260*x^2 + 100*x^3 - 10*x 
^4) + E^(2*x)*(50*x^3 - 20*x^4 + 2*x^5) + (-15*x + 10*x^2 - x^3 + E^x*(250 
 - 100*x + 10*x^2) + E^(2*x)*(50*x - 20*x^2 + 2*x^3))*Log[x] + (1250 - 500 
*x + 1300*x^2 - 500*x^3 + 50*x^4 + E^x*(-250*x^3 + 100*x^4 - 10*x^5) + (-1 
250 + 500*x - 50*x^2 + E^x*(-250*x + 100*x^2 - 10*x^3))*Log[x])*Log[(x^2 + 
 Log[x])/(2*x)])/(25*x^3 - 10*x^4 + x^5 + (25*x - 10*x^2 + x^3)*Log[x]),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(36)=72\).

Time = 41.90 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47

method	result	size
parallelrisch	\(\frac {150+10 x \,{\mathrm e}^{2 x}-100 \ln \left (\frac {\ln \left (x \right )+x^{2}}{2 x}\right ) {\mathrm e}^{x} x +250 \ln \left (\frac {\ln \left (x \right )+x^{2}}{2 x}\right )^{2} x -10 x^{2}-50 \,{\mathrm e}^{2 x}+500 \ln \left (\frac {\ln \left (x \right )+x^{2}}{2 x}\right ) {\mathrm e}^{x}-1250 \ln \left (\frac {\ln \left (x \right )+x^{2}}{2 x}\right )^{2}}{10 x -50}\)	\(94\)
risch	\(\text {Expression too large to display}\)	\(1971\)

Input:

int(((((-10*x^3+100*x^2-250*x)*exp(x)-50*x^2+500*x-1250)*ln(x)+(-10*x^5+10 
0*x^4-250*x^3)*exp(x)+50*x^4-500*x^3+1300*x^2-500*x+1250)*ln(1/2*(ln(x)+x^ 
2)/x)+((2*x^3-20*x^2+50*x)*exp(x)^2+(10*x^2-100*x+250)*exp(x)-x^3+10*x^2-1 
5*x)*ln(x)+(2*x^5-20*x^4+50*x^3)*exp(x)^2+(-10*x^4+100*x^3-260*x^2+100*x-2 
50)*exp(x)-x^5+10*x^4-15*x^3)/((x^3-10*x^2+25*x)*ln(x)+x^5-10*x^4+25*x^3), 
x,method=_RETURNVERBOSE)
 

Output:

1/10*(150+10*x*exp(x)^2-100*ln(1/2*(ln(x)+x^2)/x)*exp(x)*x+250*ln(1/2*(ln( 
x)+x^2)/x)^2*x-10*x^2-50*exp(x)^2+500*ln(1/2*(ln(x)+x^2)/x)*exp(x)-1250*ln 
(1/2*(ln(x)+x^2)/x)^2)/(-5+x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.63 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=-\frac {10 \, {\left (x - 5\right )} e^{x} \log \left (\frac {x^{2} + \log \left (x\right )}{2 \, x}\right ) - 25 \, {\left (x - 5\right )} \log \left (\frac {x^{2} + \log \left (x\right )}{2 \, x}\right )^{2} + x^{2} - {\left (x - 5\right )} e^{\left (2 \, x\right )} - 5 \, x + 10}{x - 5} \] Input:

integrate(((((-10*x^3+100*x^2-250*x)*exp(x)-50*x^2+500*x-1250)*log(x)+(-10 
*x^5+100*x^4-250*x^3)*exp(x)+50*x^4-500*x^3+1300*x^2-500*x+1250)*log(1/2*( 
log(x)+x^2)/x)+((2*x^3-20*x^2+50*x)*exp(x)^2+(10*x^2-100*x+250)*exp(x)-x^3 
+10*x^2-15*x)*log(x)+(2*x^5-20*x^4+50*x^3)*exp(x)^2+(-10*x^4+100*x^3-260*x 
^2+100*x-250)*exp(x)-x^5+10*x^4-15*x^3)/((x^3-10*x^2+25*x)*log(x)+x^5-10*x 
^4+25*x^3),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 5.92 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=- x + e^{2 x} - 10 e^{x} \log {\left (\frac {\frac {x^{2}}{2} + \frac {\log {\left (x \right )}}{2}}{x} \right )} + 25 \log {\left (\frac {\frac {x^{2}}{2} + \frac {\log {\left (x \right )}}{2}}{x} \right )}^{2} - \frac {10}{x - 5} \] Input:

integrate(((((-10*x**3+100*x**2-250*x)*exp(x)-50*x**2+500*x-1250)*ln(x)+(- 
10*x**5+100*x**4-250*x**3)*exp(x)+50*x**4-500*x**3+1300*x**2-500*x+1250)*l 
n(1/2*(ln(x)+x**2)/x)+((2*x**3-20*x**2+50*x)*exp(x)**2+(10*x**2-100*x+250) 
*exp(x)-x**3+10*x**2-15*x)*ln(x)+(2*x**5-20*x**4+50*x**3)*exp(x)**2+(-10*x 
**4+100*x**3-260*x**2+100*x-250)*exp(x)-x**5+10*x**4-15*x**3)/((x**3-10*x* 
*2+25*x)*ln(x)+x**5-10*x**4+25*x**3),x)
 

Output:

-x + exp(2*x) - 10*exp(x)*log((x**2/2 + log(x)/2)/x) + 25*log((x**2/2 + lo 
g(x)/2)/x)**2 - 10/(x - 5)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (32) = 64\).

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.92 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=\frac {25 \, {\left (x - 5\right )} \log \left (x^{2} + \log \left (x\right )\right )^{2} + 25 \, {\left (x - 5\right )} \log \left (x\right )^{2} - x^{2} + {\left (x - 5\right )} e^{\left (2 \, x\right )} + 10 \, {\left (x \log \left (2\right ) + {\left (x - 5\right )} \log \left (x\right ) - 5 \, \log \left (2\right )\right )} e^{x} - 10 \, {\left ({\left (x - 5\right )} e^{x} + 5 \, x \log \left (2\right ) + 5 \, {\left (x - 5\right )} \log \left (x\right ) - 25 \, \log \left (2\right )\right )} \log \left (x^{2} + \log \left (x\right )\right ) + 50 \, {\left (x \log \left (2\right ) - 5 \, \log \left (2\right )\right )} \log \left (x\right ) + 5 \, x - 10}{x - 5} \] Input:

integrate(((((-10*x^3+100*x^2-250*x)*exp(x)-50*x^2+500*x-1250)*log(x)+(-10 
*x^5+100*x^4-250*x^3)*exp(x)+50*x^4-500*x^3+1300*x^2-500*x+1250)*log(1/2*( 
log(x)+x^2)/x)+((2*x^3-20*x^2+50*x)*exp(x)^2+(10*x^2-100*x+250)*exp(x)-x^3 
+10*x^2-15*x)*log(x)+(2*x^5-20*x^4+50*x^3)*exp(x)^2+(-10*x^4+100*x^3-260*x 
^2+100*x-250)*exp(x)-x^5+10*x^4-15*x^3)/((x^3-10*x^2+25*x)*log(x)+x^5-10*x 
^4+25*x^3),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (32) = 64\).

Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 4.53 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=\frac {10 \, x e^{x} \log \left (2\right ) - 10 \, x e^{x} \log \left (x^{2} + \log \left (x\right )\right ) - 50 \, x \log \left (2\right ) \log \left (x^{2} + \log \left (x\right )\right ) + 25 \, x \log \left (x^{2} + \log \left (x\right )\right )^{2} + 10 \, x e^{x} \log \left (x\right ) + 50 \, x \log \left (2\right ) \log \left (x\right ) - 50 \, x \log \left (x^{2} + \log \left (x\right )\right ) \log \left (x\right ) + 25 \, x \log \left (x\right )^{2} - x^{2} + x e^{\left (2 \, x\right )} - 50 \, e^{x} \log \left (2\right ) + 50 \, e^{x} \log \left (x^{2} + \log \left (x\right )\right ) + 250 \, \log \left (2\right ) \log \left (x^{2} + \log \left (x\right )\right ) - 125 \, \log \left (x^{2} + \log \left (x\right )\right )^{2} - 50 \, e^{x} \log \left (x\right ) - 250 \, \log \left (2\right ) \log \left (x\right ) + 250 \, \log \left (x^{2} + \log \left (x\right )\right ) \log \left (x\right ) - 125 \, \log \left (x\right )^{2} + 5 \, x - 5 \, e^{\left (2 \, x\right )} - 10}{x - 5} \] Input:

integrate(((((-10*x^3+100*x^2-250*x)*exp(x)-50*x^2+500*x-1250)*log(x)+(-10 
*x^5+100*x^4-250*x^3)*exp(x)+50*x^4-500*x^3+1300*x^2-500*x+1250)*log(1/2*( 
log(x)+x^2)/x)+((2*x^3-20*x^2+50*x)*exp(x)^2+(10*x^2-100*x+250)*exp(x)-x^3 
+10*x^2-15*x)*log(x)+(2*x^5-20*x^4+50*x^3)*exp(x)^2+(-10*x^4+100*x^3-260*x 
^2+100*x-250)*exp(x)-x^5+10*x^4-15*x^3)/((x^3-10*x^2+25*x)*log(x)+x^5-10*x 
^4+25*x^3),x, algorithm="giac")
 

Output:

(10*x*e^x*log(2) - 10*x*e^x*log(x^2 + log(x)) - 50*x*log(2)*log(x^2 + log( 
x)) + 25*x*log(x^2 + log(x))^2 + 10*x*e^x*log(x) + 50*x*log(2)*log(x) - 50 
*x*log(x^2 + log(x))*log(x) + 25*x*log(x)^2 - x^2 + x*e^(2*x) - 50*e^x*log 
(2) + 50*e^x*log(x^2 + log(x)) + 250*log(2)*log(x^2 + log(x)) - 125*log(x^ 
2 + log(x))^2 - 50*e^x*log(x) - 250*log(2)*log(x) + 250*log(x^2 + log(x))* 
log(x) - 125*log(x)^2 + 5*x - 5*e^(2*x) - 10)/(x - 5)
 

Mupad [B] (verification not implemented)

Time = 2.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx={\mathrm {e}}^{2\,x}-x-\frac {10}{x-5}-10\,{\mathrm {e}}^x\,\ln \left (\frac {\frac {\ln \left (x\right )}{2}+\frac {x^2}{2}}{x}\right )+25\,{\ln \left (\frac {\frac {\ln \left (x\right )}{2}+\frac {x^2}{2}}{x}\right )}^2 \] Input:

int(-(exp(x)*(260*x^2 - 100*x - 100*x^3 + 10*x^4 + 250) + log((log(x)/2 + 
x^2/2)/x)*(500*x + exp(x)*(250*x^3 - 100*x^4 + 10*x^5) + log(x)*(50*x^2 - 
500*x + exp(x)*(250*x - 100*x^2 + 10*x^3) + 1250) - 1300*x^2 + 500*x^3 - 5 
0*x^4 - 1250) - exp(2*x)*(50*x^3 - 20*x^4 + 2*x^5) - log(x)*(exp(2*x)*(50* 
x - 20*x^2 + 2*x^3) - 15*x + exp(x)*(10*x^2 - 100*x + 250) + 10*x^2 - x^3) 
 + 15*x^3 - 10*x^4 + x^5)/(log(x)*(25*x - 10*x^2 + x^3) + 25*x^3 - 10*x^4 
+ x^5),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.55 \[ \int \frac {-15 x^3+10 x^4-x^5+e^x \left (-250+100 x-260 x^2+100 x^3-10 x^4\right )+e^{2 x} \left (50 x^3-20 x^4+2 x^5\right )+\left (-15 x+10 x^2-x^3+e^x \left (250-100 x+10 x^2\right )+e^{2 x} \left (50 x-20 x^2+2 x^3\right )\right ) \log (x)+\left (1250-500 x+1300 x^2-500 x^3+50 x^4+e^x \left (-250 x^3+100 x^4-10 x^5\right )+\left (-1250+500 x-50 x^2+e^x \left (-250 x+100 x^2-10 x^3\right )\right ) \log (x)\right ) \log \left (\frac {x^2+\log (x)}{2 x}\right )}{25 x^3-10 x^4+x^5+\left (25 x-10 x^2+x^3\right ) \log (x)} \, dx=\frac {e^{2 x} x -5 e^{2 x}-10 e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}}{2 x}\right ) x +50 e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}}{2 x}\right )+25 \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}}{2 x}\right )^{2} x -125 \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}}{2 x}\right )^{2}-x^{2}+3 x}{-5+x} \] Input:

int(((((-10*x^3+100*x^2-250*x)*exp(x)-50*x^2+500*x-1250)*log(x)+(-10*x^5+1 
00*x^4-250*x^3)*exp(x)+50*x^4-500*x^3+1300*x^2-500*x+1250)*log(1/2*(log(x) 
+x^2)/x)+((2*x^3-20*x^2+50*x)*exp(x)^2+(10*x^2-100*x+250)*exp(x)-x^3+10*x^ 
2-15*x)*log(x)+(2*x^5-20*x^4+50*x^3)*exp(x)^2+(-10*x^4+100*x^3-260*x^2+100 
*x-250)*exp(x)-x^5+10*x^4-15*x^3)/((x^3-10*x^2+25*x)*log(x)+x^5-10*x^4+25* 
x^3),x)
 

Output:

(e**(2*x)*x - 5*e**(2*x) - 10*e**x*log((log(x) + x**2)/(2*x))*x + 50*e**x* 
log((log(x) + x**2)/(2*x)) + 25*log((log(x) + x**2)/(2*x))**2*x - 125*log( 
(log(x) + x**2)/(2*x))**2 - x**2 + 3*x)/(x - 5)