\(\int \frac {e^x (-625 x+625 x^2)+e^{\frac {e^{-x} (-4+e^x (2 x+x^2))}{x}} (2500+2500 x+625 e^x x^2)}{e^{x+\frac {2 e^{-x} (-4+e^x (2 x+x^2))}{x}} x^2+e^x (4 x^2+4 x^3+x^4)+e^x (-4 x^2-2 x^3) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} (-4+e^x (2 x+x^2))}{x}} (e^x (4 x^2+2 x^3)-2 e^x x^2 \log (5 x))} \, dx\) [1011]

Optimal result

Integrand size = 190, antiderivative size = 30 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=\frac {625}{-2-e^{2-\frac {4 e^{-x}}{x}+x}-x+\log (5 x)} \] Output:

625/(ln(5*x)-exp(x+2-4/exp(x)/x)-2-x)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=-\frac {625 e^{\frac {4 e^{-x}}{x}}}{e^{2+x}+e^{\frac {4 e^{-x}}{x}} (2+x)-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \] Input:

Integrate[(E^x*(-625*x + 625*x^2) + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(25 
00 + 2500*x + 625*E^x*x^2))/(E^(x + (2*(-4 + E^x*(2*x + x^2)))/(E^x*x))*x^ 
2 + E^x*(4*x^2 + 4*x^3 + x^4) + E^x*(-4*x^2 - 2*x^3)*Log[5*x] + E^x*x^2*Lo 
g[5*x]^2 + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(E^x*(4*x^2 + 2*x^3) - 2*E^x 
*x^2*Log[5*x])),x]
 

Output:

(-625*E^(4/(E^x*x)))/(E^(2 + x) + E^(4/(E^x*x))*(2 + x) - E^(4/(E^x*x))*Lo 
g[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.

\(\displaystyle \int \frac {e^x \left (625 x^2-625 x\right )+e^{\frac {e^{-x} \left (e^x \left (x^2+2 x\right )-4\right )}{x}} \left (625 e^x x^2+2500 x+2500\right )}{e^{\frac {2 e^{-x} \left (e^x \left (x^2+2 x\right )-4\right )}{x}+x} x^2+e^x x^2 \log ^2(5 x)+e^x \left (-2 x^3-4 x^2\right ) \log (5 x)+e^{\frac {e^{-x} \left (e^x \left (x^2+2 x\right )-4\right )}{x}} \left (e^x \left (2 x^3+4 x^2\right )-2 e^x x^2 \log (5 x)\right )+e^x \left (x^4+4 x^3+4 x^2\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {625 e^{\frac {4 e^{-x}}{x}} \left (e^{x+2} x^2+e^{\frac {4 e^{-x}}{x}} (x-1) x+4 e^2 (x+1)\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} (x+2)+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 625 \int -\frac {e^{\frac {4 e^{-x}}{x}} \left (-e^{x+2} x^2+e^{\frac {4 e^{-x}}{x}} (1-x) x-4 e^2 (x+1)\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} (x+2)+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -625 \int \frac {e^{\frac {4 e^{-x}}{x}} \left (-e^{x+2} x^2+e^{\frac {4 e^{-x}}{x}} (1-x) x-4 e^2 (x+1)\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} (x+2)+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -625 \int \left (\frac {e^{\frac {4 e^{-x}}{x}} \left (e^{\frac {4 e^{-x}}{x}} x^3+e^{\frac {4 e^{-x}}{x}} x^2-e^{\frac {4 e^{-x}}{x}} \log (5 x) x^2+e^{\frac {4 e^{-x}}{x}} x-4 e^2 x-4 e^2\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}-\frac {e^{\frac {4 e^{-x}}{x}}}{e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -625 \int \frac {e^{\frac {4 e^{-x}}{x}} \left (-e^{x+2} x^2-e^{\frac {4 e^{-x}}{x}} (x-1) x-4 e^2 (x+1)\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} (x+2)+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -625 \int \left (\frac {e^{\frac {4 e^{-x}}{x}} \left (e^{\frac {4 e^{-x}}{x}} x^3+e^{\frac {4 e^{-x}}{x}} x^2-e^{\frac {4 e^{-x}}{x}} \log (5 x) x^2+e^{\frac {4 e^{-x}}{x}} x-4 e^2 x-4 e^2\right )}{x^2 \left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}-\frac {e^{\frac {4 e^{-x}}{x}}}{e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -625 \left (-4 \int \frac {e^{2+\frac {4 e^{-x}}{x}}}{x^2 \left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx+\int \frac {e^{\frac {8 e^{-x}}{x}}}{\left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx-4 \int \frac {e^{2+\frac {4 e^{-x}}{x}}}{x \left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx+\int \frac {e^{\frac {8 e^{-x}}{x}}}{x \left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx+\int \frac {e^{\frac {8 e^{-x}}{x}} x}{\left (e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx-\int \frac {e^{\frac {4 e^{-x}}{x}}}{e^{\frac {4 e^{-x}}{x}} x+2 e^{\frac {4 e^{-x}}{x}}+e^{x+2}-e^{\frac {4 e^{-x}}{x}} \log (5 x)}dx-\int \frac {e^{\frac {8 e^{-x}}{x}} \log (5 x)}{\left (-e^{\frac {4 e^{-x}}{x}} x-2 e^{\frac {4 e^{-x}}{x}}-e^{x+2}+e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}dx\right )\)

Input:

Int[(E^x*(-625*x + 625*x^2) + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(2500 + 2 
500*x + 625*E^x*x^2))/(E^(x + (2*(-4 + E^x*(2*x + x^2)))/(E^x*x))*x^2 + E^ 
x*(4*x^2 + 4*x^3 + x^4) + E^x*(-4*x^2 - 2*x^3)*Log[5*x] + E^x*x^2*Log[5*x] 
^2 + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(E^x*(4*x^2 + 2*x^3) - 2*E^x*x^2*L 
og[5*x])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17

method	result	size
parallelrisch	\(-\frac {625}{x -\ln \left (5 x \right )+{\mathrm e}^{\frac {\left (\left (x^{2}+2 x \right ) {\mathrm e}^{x}-4\right ) {\mathrm e}^{-x}}{x}}+2}\)	\(35\)
risch	\(-\frac {625}{x -\ln \left (5 x \right )+{\mathrm e}^{\frac {\left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x -4\right ) {\mathrm e}^{-x}}{x}}+2}\)	\(36\)

Input:

int(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625* 
x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^ 
2*exp(x)*ln(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+ 
x^2*exp(x)*ln(5*x)^2+(-2*x^3-4*x^2)*exp(x)*ln(5*x)+(x^4+4*x^3+4*x^2)*exp(x 
)),x,method=_RETURNVERBOSE)
 

Output:

-625/(x-ln(5*x)+exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=-\frac {625}{x + e^{\left (\frac {{\left ({\left (x^{2} + 2 \, x\right )} e^{x} - 4\right )} e^{\left (-x\right )}}{x}\right )} - \log \left (5 \, x\right ) + 2} \] Input:

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x) 
+(625*x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+ 
(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp 
(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x 
^2)*exp(x)),x, algorithm="fricas")
 

Output:

-625/(x + e^(((x^2 + 2*x)*e^x - 4)*e^(-x)/x) - log(5*x) + 2)
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=- \frac {625}{x + e^{\frac {\left (\left (x^{2} + 2 x\right ) e^{x} - 4\right ) e^{- x}}{x}} - \log {\left (5 x \right )} + 2} \] Input:

integrate(((625*exp(x)*x**2+2500*x+2500)*exp(((x**2+2*x)*exp(x)-4)/exp(x)/ 
x)+(625*x**2-625*x)*exp(x))/(x**2*exp(x)*exp(((x**2+2*x)*exp(x)-4)/exp(x)/ 
x)**2+(-2*x**2*exp(x)*ln(5*x)+(2*x**3+4*x**2)*exp(x))*exp(((x**2+2*x)*exp( 
x)-4)/exp(x)/x)+x**2*exp(x)*ln(5*x)**2+(-2*x**3-4*x**2)*exp(x)*ln(5*x)+(x* 
*4+4*x**3+4*x**2)*exp(x)),x)
 

Output:

-625/(x + exp(((x**2 + 2*x)*exp(x) - 4)*exp(-x)/x) - log(5*x) + 2)
 

Maxima [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=-\frac {625 \, e^{\left (\frac {4 \, e^{\left (-x\right )}}{x}\right )}}{{\left (x - \log \left (5\right ) - \log \left (x\right ) + 2\right )} e^{\left (\frac {4 \, e^{\left (-x\right )}}{x}\right )} + e^{\left (x + 2\right )}} \] Input:

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x) 
+(625*x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+ 
(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp 
(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x 
^2)*exp(x)),x, algorithm="maxima")
 

Output:

-625*e^(4*e^(-x)/x)/((x - log(5) - log(x) + 2)*e^(4*e^(-x)/x) + e^(x + 2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16446 vs. \(2 (26) = 52\).

Time = 0.39 (sec) , antiderivative size = 16446, normalized size of antiderivative = 548.20 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=\text {Too large to display} \] Input:

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x) 
+(625*x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+ 
(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp 
(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x 
^2)*exp(x)),x, algorithm="giac")
 

Output:

-625*(x^7*e^(2*x + (x^2 + 4*e^(-x))/x) - 3*x^6*e^(2*x + (x^2 + 4*e^(-x))/x 
)*log(5) + 3*x^5*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5)^2 - x^4*e^(2*x + (x^2 
 + 4*e^(-x))/x)*log(5)^3 - 3*x^6*e^(2*x + (x^2 + 4*e^(-x))/x)*log(x) + 6*x 
^5*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5)*log(x) - 3*x^4*e^(2*x + (x^2 + 4*e^ 
(-x))/x)*log(5)^2*log(x) + 3*x^5*e^(2*x + (x^2 + 4*e^(-x))/x)*log(x)^2 - 3 
*x^4*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5)*log(x)^2 - x^4*e^(2*x + (x^2 + 4* 
e^(-x))/x)*log(x)^3 + 4*x^6*e^(2*x + (x^2 + 4*e^(-x))/x) + x^6*e^(x + 2*(x 
^2 + x - 2*e^(-x))/x + (x^2 + 4*e^(-x))/x) + 8*x^6*e^(x + (x^2 + 4*e^(-x)) 
/x) - 8*x^5*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5) - 2*x^5*e^(x + 2*(x^2 + x 
- 2*e^(-x))/x + (x^2 + 4*e^(-x))/x)*log(5) - 24*x^5*e^(x + (x^2 + 4*e^(-x) 
)/x)*log(5) + 4*x^4*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5)^2 + x^4*e^(x + 2*( 
x^2 + x - 2*e^(-x))/x + (x^2 + 4*e^(-x))/x)*log(5)^2 + 24*x^4*e^(x + (x^2 
+ 4*e^(-x))/x)*log(5)^2 - 8*x^3*e^(x + (x^2 + 4*e^(-x))/x)*log(5)^3 - 8*x^ 
5*e^(2*x + (x^2 + 4*e^(-x))/x)*log(x) - 2*x^5*e^(x + 2*(x^2 + x - 2*e^(-x) 
)/x + (x^2 + 4*e^(-x))/x)*log(x) - 24*x^5*e^(x + (x^2 + 4*e^(-x))/x)*log(x 
) + 8*x^4*e^(2*x + (x^2 + 4*e^(-x))/x)*log(5)*log(x) + 2*x^4*e^(x + 2*(x^2 
 + x - 2*e^(-x))/x + (x^2 + 4*e^(-x))/x)*log(5)*log(x) + 48*x^4*e^(x + (x^ 
2 + 4*e^(-x))/x)*log(5)*log(x) - 24*x^3*e^(x + (x^2 + 4*e^(-x))/x)*log(5)^ 
2*log(x) + 4*x^4*e^(2*x + (x^2 + 4*e^(-x))/x)*log(x)^2 + x^4*e^(x + 2*(x^2 
 + x - 2*e^(-x))/x + (x^2 + 4*e^(-x))/x)*log(x)^2 + 24*x^4*e^(x + (x^2 ...
 

Mupad [B] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=-\frac {625}{x-\ln \left (5\right )-\ln \left (x\right )+{\mathrm {e}}^2\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x+2} \] Input:

int((exp((exp(-x)*(exp(x)*(2*x + x^2) - 4))/x)*(2500*x + 625*x^2*exp(x) + 
2500) - exp(x)*(625*x - 625*x^2))/(exp((exp(-x)*(exp(x)*(2*x + x^2) - 4))/ 
x)*(exp(x)*(4*x^2 + 2*x^3) - 2*x^2*log(5*x)*exp(x)) + exp(x)*(4*x^2 + 4*x^ 
3 + x^4) - log(5*x)*exp(x)*(4*x^2 + 2*x^3) + x^2*exp((2*exp(-x)*(exp(x)*(2 
*x + x^2) - 4))/x)*exp(x) + x^2*log(5*x)^2*exp(x)),x)
 

Output:

-625/(x - log(5) - log(x) + exp(2)*exp(-(4*exp(-x))/x)*exp(x) + 2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx=\frac {625 e^{\frac {4}{e^{x} x}}}{e^{\frac {4}{e^{x} x}} \mathrm {log}\left (5 x \right )-e^{\frac {4}{e^{x} x}} x -2 e^{\frac {4}{e^{x} x}}-e^{x} e^{2}} \] Input:

int(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625* 
x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^ 
2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x) 
+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x^2)*ex 
p(x)),x)
 

Output:

(625*e**(4/(e**x*x)))/(e**(4/(e**x*x))*log(5*x) - e**(4/(e**x*x))*x - 2*e* 
*(4/(e**x*x)) - e**x*e**2)