\(\int \frac {40 x+e^{e^{5/x}+x} (-20 e^{5/x}+8 x+4 x^2)+e^{25 x^2-50 x^3+25 x^4} (-4-200 x^2+600 x^3-400 x^4)}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4)} \, dx\) [755]

Optimal result

Integrand size = 184, antiderivative size = 36 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=\frac {4}{x \left (e^{25 (-1+x)^2 x^2}-\left (5+e^{e^{5/x}+x}\right ) x\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=-\frac {4 e^{50 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )} \] Input:

Integrate[(40*x + E^(E^(5/x) + x)*(-20*E^(5/x) + 8*x + 4*x^2) + E^(25*x^2 
- 50*x^3 + 25*x^4)*(-4 - 200*x^2 + 600*x^3 - 400*x^4))/(E^(50*x^2 - 100*x^ 
3 + 50*x^4)*x^2 - 10*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 25*x^4 + E^(2*E^(5 
/x) + 2*x)*x^4 + E^(E^(5/x) + x)*(-2*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 10 
*x^4)),x]
 

Output:

(-4*E^(50*x^3))/(x*(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x 
 + 50*x^3)*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.

\(\displaystyle \int \frac {e^{x+e^{5/x}} \left (4 x^2+8 x-20 e^{5/x}\right )+e^{25 x^4-50 x^3+25 x^2} \left (-400 x^4+600 x^3-200 x^2-4\right )+40 x}{e^{2 x+2 e^{5/x}} x^4+25 x^4-10 e^{25 x^4-50 x^3+25 x^2} x^3+e^{50 x^4-100 x^3+50 x^2} x^2+e^{x+e^{5/x}} \left (10 x^4-2 e^{25 x^4-50 x^3+25 x^2} x^3\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{100 x^3} \left (e^{x+e^{5/x}} \left (4 x^2+8 x-20 e^{5/x}\right )+e^{25 x^4-50 x^3+25 x^2} \left (-400 x^4+600 x^3-200 x^2-4\right )+40 x\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 x^4+25 x^2}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}-\frac {4 e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 e^{50 x^3} \left (10 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} (x+2) x-5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}-e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int -\frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^{50 x^3} \left (-10 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} (x+2) x+5 e^{50 x^3+x+e^{5/x}+\frac {5}{x}}+e^{25 \left (x^4+x^2\right )} \left (100 x^4-150 x^3+50 x^2+1\right )\right )}{x^2 \left (-5 e^{50 x^3} x-e^{50 x^3+x+e^{5/x}} x+e^{25 \left (x^4+x^2\right )}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^{100 x^3} \left (100 e^{x+e^{5/x}} x^5+500 x^5-150 e^{x+e^{5/x}} x^4-750 x^4+50 e^{x+e^{5/x}} x^3+250 x^3-e^{x+e^{5/x}} x^2-e^{x+e^{5/x}} x-5 x+5 e^{x+e^{5/x}+\frac {5}{x}}\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )^2}-\frac {e^{50 x^3} \left (100 x^4-150 x^3+50 x^2+1\right )}{x^2 \left (5 e^{50 x^3} x+e^{50 x^3+x+e^{5/x}} x-e^{25 x^4+25 x^2}\right )}\right )dx\)

Input:

Int[(40*x + E^(E^(5/x) + x)*(-20*E^(5/x) + 8*x + 4*x^2) + E^(25*x^2 - 50*x 
^3 + 25*x^4)*(-4 - 200*x^2 + 600*x^3 - 400*x^4))/(E^(50*x^2 - 100*x^3 + 50 
*x^4)*x^2 - 10*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 25*x^4 + E^(2*E^(5/x) + 
2*x)*x^4 + E^(E^(5/x) + x)*(-2*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 10*x^4)) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.65 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {4}{x \left (x \,{\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}-{\mathrm e}^{25 x^{2} \left (-1+x \right )^{2}}+5 x \right )}\)	\(36\)
parallelrisch	\(-\frac {4}{x \left (x \,{\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}-{\mathrm e}^{25 x^{4}-50 x^{3}+25 x^{2}}+5 x \right )}\)	\(42\)

Input:

int(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4) 
*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4 
-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2)^2-10 
*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x,method=_RETURNVERBOSE)
 

Output:

-4/x/(x*exp(exp(5/x)+x)-exp(25*x^2*(-1+x)^2)+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=-\frac {4}{x^{2} e^{\left (x + e^{\frac {5}{x}}\right )} + 5 \, x^{2} - x e^{\left (25 \, x^{4} - 50 \, x^{3} + 25 \, x^{2}\right )}} \] Input:

integrate(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200* 
x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp( 
25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2 
)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="fricas")
 

Output:

-4/(x^2*e^(x + e^(5/x)) + 5*x^2 - x*e^(25*x^4 - 50*x^3 + 25*x^2))
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=- \frac {4}{x^{2} e^{x + e^{\frac {5}{x}}} + 5 x^{2} - x e^{25 x^{4} - 50 x^{3} + 25 x^{2}}} \] Input:

integrate(((-20*exp(5/x)+4*x**2+8*x)*exp(exp(5/x)+x)+(-400*x**4+600*x**3-2 
00*x**2-4)*exp(25*x**4-50*x**3+25*x**2)+40*x)/(x**4*exp(exp(5/x)+x)**2+(-2 
*x**3*exp(25*x**4-50*x**3+25*x**2)+10*x**4)*exp(exp(5/x)+x)+x**2*exp(25*x* 
*4-50*x**3+25*x**2)**2-10*x**3*exp(25*x**4-50*x**3+25*x**2)+25*x**4),x)
 

Output:

-4/(x**2*exp(x + exp(5/x)) + 5*x**2 - x*exp(25*x**4 - 50*x**3 + 25*x**2))
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=-\frac {4 \, e^{\left (50 \, x^{3}\right )}}{5 \, x^{2} e^{\left (50 \, x^{3}\right )} + x^{2} e^{\left (50 \, x^{3} + x + e^{\frac {5}{x}}\right )} - x e^{\left (25 \, x^{4} + 25 \, x^{2}\right )}} \] Input:

integrate(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200* 
x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp( 
25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2 
)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="maxima")
 

Output:

-4*e^(50*x^3)/(5*x^2*e^(50*x^3) + x^2*e^(50*x^3 + x + e^(5/x)) - x*e^(25*x 
^4 + 25*x^2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14225 vs. \(2 (34) = 68\).

Time = 0.32 (sec) , antiderivative size = 14225, normalized size of antiderivative = 395.14 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200* 
x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp( 
25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2 
)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="giac")
 

Output:

-4*(10000*x^11*e^(x + 2*(x^2 + x*e^(5/x) + 5)/x + e^(5/x)) + 100000*x^11*e 
^(x + (x^2 + x*e^(5/x) + 5)/x + 5/x + e^(5/x)) + 250000*x^11*e^(x + 10/x + 
 e^(5/x)) + 50000*x^11*e^(2*(x^2 + x*e^(5/x) + 5)/x) + 500000*x^11*e^((x^2 
 + x*e^(5/x) + 5)/x + 5/x) + 1250000*x^11*e^(10/x) - 10000*x^10*e^(25*x^4 
- 50*x^3 + 25*x^2 + x + (x^2 + x*e^(5/x) + 5)/x + 5/x + e^(5/x)) - 50000*x 
^10*e^(25*x^4 - 50*x^3 + 25*x^2 + x + 10/x + e^(5/x)) - 50000*x^10*e^(25*x 
^4 - 50*x^3 + 25*x^2 + (x^2 + x*e^(5/x) + 5)/x + 5/x) - 250000*x^10*e^(25* 
x^4 - 50*x^3 + 25*x^2 + 10/x) - 30000*x^10*e^(x + 2*(x^2 + x*e^(5/x) + 5)/ 
x + e^(5/x)) - 300000*x^10*e^(x + (x^2 + x*e^(5/x) + 5)/x + 5/x + e^(5/x)) 
 - 750000*x^10*e^(x + 10/x + e^(5/x)) - 150000*x^10*e^(2*(x^2 + x*e^(5/x) 
+ 5)/x) - 1500000*x^10*e^((x^2 + x*e^(5/x) + 5)/x + 5/x) - 3750000*x^10*e^ 
(10/x) + 30000*x^9*e^(25*x^4 - 50*x^3 + 25*x^2 + x + (x^2 + x*e^(5/x) + 5) 
/x + 5/x + e^(5/x)) + 150000*x^9*e^(25*x^4 - 50*x^3 + 25*x^2 + x + 10/x + 
e^(5/x)) + 150000*x^9*e^(25*x^4 - 50*x^3 + 25*x^2 + (x^2 + x*e^(5/x) + 5)/ 
x + 5/x) + 750000*x^9*e^(25*x^4 - 50*x^3 + 25*x^2 + 10/x) + 32500*x^9*e^(x 
 + 2*(x^2 + x*e^(5/x) + 5)/x + e^(5/x)) + 325000*x^9*e^(x + (x^2 + x*e^(5/ 
x) + 5)/x + 5/x + e^(5/x)) + 812500*x^9*e^(x + 10/x + e^(5/x)) + 162500*x^ 
9*e^(2*(x^2 + x*e^(5/x) + 5)/x) + 1625000*x^9*e^((x^2 + x*e^(5/x) + 5)/x + 
 5/x) + 4062500*x^9*e^(10/x) - 32500*x^8*e^(25*x^4 - 50*x^3 + 25*x^2 + x + 
 (x^2 + x*e^(5/x) + 5)/x + 5/x + e^(5/x)) - 162500*x^8*e^(25*x^4 - 50*x...
 

Mupad [B] (verification not implemented)

Time = 6.47 (sec) , antiderivative size = 274, normalized size of antiderivative = 7.61 \[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=-\frac {20\,{\mathrm {e}}^{\frac {5}{x}+25\,x^2-50\,x^3+25\,x^4}-100\,x\,{\mathrm {e}}^{5/x}+20\,x^3-{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}\,\left (-400\,x^5+600\,x^4-200\,x^3+4\,x^2+4\,x\right )}{\left (5\,x-{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+x\,{\mathrm {e}}^{x+{\mathrm {e}}^{5/x}}\right )\,\left (5\,x\,{\mathrm {e}}^{\frac {5}{x}+25\,x^2-50\,x^3+25\,x^4}-x^2\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-x^3\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+50\,x^4\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-150\,x^5\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+100\,x^6\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-25\,x^2\,{\mathrm {e}}^{5/x}+5\,x^4\right )} \] Input:

int((40*x + exp(x + exp(5/x))*(8*x - 20*exp(5/x) + 4*x^2) - exp(25*x^2 - 5 
0*x^3 + 25*x^4)*(200*x^2 - 600*x^3 + 400*x^4 + 4))/(x^2*exp(50*x^2 - 100*x 
^3 + 50*x^4) - 10*x^3*exp(25*x^2 - 50*x^3 + 25*x^4) - exp(x + exp(5/x))*(2 
*x^3*exp(25*x^2 - 50*x^3 + 25*x^4) - 10*x^4) + x^4*exp(2*x + 2*exp(5/x)) + 
 25*x^4),x)
 

Output:

-(20*exp(5/x + 25*x^2 - 50*x^3 + 25*x^4) - 100*x*exp(5/x) + 20*x^3 - exp(2 
5*x^2 - 50*x^3 + 25*x^4)*(4*x + 4*x^2 - 200*x^3 + 600*x^4 - 400*x^5))/((5* 
x - exp(25*x^2 - 50*x^3 + 25*x^4) + x*exp(x + exp(5/x)))*(5*x*exp(5/x + 25 
*x^2 - 50*x^3 + 25*x^4) - x^2*exp(25*x^2 - 50*x^3 + 25*x^4) - x^3*exp(25*x 
^2 - 50*x^3 + 25*x^4) + 50*x^4*exp(25*x^2 - 50*x^3 + 25*x^4) - 150*x^5*exp 
(25*x^2 - 50*x^3 + 25*x^4) + 100*x^6*exp(25*x^2 - 50*x^3 + 25*x^4) - 25*x^ 
2*exp(5/x) + 5*x^4))
 

Reduce [F]

\[ \int \frac {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx=\int \frac {\left (-20 \,{\mathrm e}^{\frac {5}{x}}+4 x^{2}+8 x \right ) {\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}+\left (-400 x^{4}+600 x^{3}-200 x^{2}-4\right ) {\mathrm e}^{25 x^{4}-50 x^{3}+25 x^{2}}+40 x}{x^{4} \left ({\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}\right )^{2}+\left (-2 x^{3} {\mathrm e}^{25 x^{4}-50 x^{3}+25 x^{2}}+10 x^{4}\right ) {\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}+x^{2} \left ({\mathrm e}^{25 x^{4}-50 x^{3}+25 x^{2}}\right )^{2}-10 x^{3} {\mathrm e}^{25 x^{4}-50 x^{3}+25 x^{2}}+25 x^{4}}d x \] Input:

int(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4) 
*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4 
-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2)^2-10 
*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x)
 

Output:

int(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4) 
*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4 
-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2)^2-10 
*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x)