\(\int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 (40 x-40 x^2)+e^8 (16-32 x+16 x^2)+e^6 (400-800 x+400 x^2)+e^{2 x} (25 e^6 x^2+10 e^7 x^2+e^8 x^2)+e^4 (8 x-8 x^2+e^3 (160-320 x+160 x^2))+e^x (-10 e^3 x^2+e^8 (-8 x+8 x^2)+e^6 (-200 x+200 x^2)+e^4 (-2 x^2+e^3 (-80 x+80 x^2)))}} (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3))}{-x^3+e^6 (-1200 x+2400 x^2-1200 x^3)+e^3 (-60 x^2+60 x^3)+e^{12} (-64+192 x-192 x^2+64 x^3)+e^9 (-8000+24000 x-24000 x^2+8000 x^3)+e^{3 x} (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3)+e^8 (-48 x+96 x^2-48 x^3+e^3 (-960+2880 x-2880 x^2+960 x^3))+e^4 (-12 x^2+12 x^3+e^3 (-480 x+960 x^2-480 x^3)+e^6 (-4800+14400 x-14400 x^2+4800 x^3))+e^{2 x} (-75 e^6 x^3+e^{12} (-12 x^2+12 x^3)+e^9 (-1500 x^2+1500 x^3)+e^8 (-3 x^3+e^3 (-180 x^2+180 x^3))+e^4 (-30 e^3 x^3+e^6 (-900 x^2+900 x^3)))+e^x (15 e^3 x^3+e^6 (600 x^2-600 x^3)+e^{12} (48 x-96 x^2+48 x^3)+e^9 (6000 x-12000 x^2+6000 x^3)+e^8 (24 x^2-24 x^3+e^3 (720 x-1440 x^2+720 x^3))+e^4 (3 x^3+e^3 (240 x^2-240 x^3)+e^6 (3600 x-7200 x^2+3600 x^3)))} \, dx\) [2261]

Optimal result

Integrand size = 701, antiderivative size = 31 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=e^{\frac {2}{\left (4+e^x-\frac {4+\frac {x}{5 e^3+e^4}}{x}\right )^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=e^{\frac {2 e^6 (5+e)^2 x^2}{\left (20 e^3 (-1+x)+4 e^4 (-1+x)-x+5 e^{3+x} x+e^{4+x} x\right )^2}} \] Input:

Integrate[(E^((50*E^6*x^2 + 20*E^7*x^2 + 2*E^8*x^2)/(x^2 + E^3*(40*x - 40* 
x^2) + E^8*(16 - 32*x + 16*x^2) + E^6*(400 - 800*x + 400*x^2) + E^(2*x)*(2 
5*E^6*x^2 + 10*E^7*x^2 + E^8*x^2) + E^4*(8*x - 8*x^2 + E^3*(160 - 320*x + 
160*x^2)) + E^x*(-10*E^3*x^2 + E^8*(-8*x + 8*x^2) + E^6*(-200*x + 200*x^2) 
 + E^4*(-2*x^2 + E^3*(-80*x + 80*x^2)))))*(-2000*E^9*x - 1200*E^10*x - 240 
*E^11*x - 16*E^12*x + E^x*(-500*E^9*x^3 - 300*E^10*x^3 - 60*E^11*x^3 - 4*E 
^12*x^3)))/(-x^3 + E^6*(-1200*x + 2400*x^2 - 1200*x^3) + E^3*(-60*x^2 + 60 
*x^3) + E^12*(-64 + 192*x - 192*x^2 + 64*x^3) + E^9*(-8000 + 24000*x - 240 
00*x^2 + 8000*x^3) + E^(3*x)*(125*E^9*x^3 + 75*E^10*x^3 + 15*E^11*x^3 + E^ 
12*x^3) + E^8*(-48*x + 96*x^2 - 48*x^3 + E^3*(-960 + 2880*x - 2880*x^2 + 9 
60*x^3)) + E^4*(-12*x^2 + 12*x^3 + E^3*(-480*x + 960*x^2 - 480*x^3) + E^6* 
(-4800 + 14400*x - 14400*x^2 + 4800*x^3)) + E^(2*x)*(-75*E^6*x^3 + E^12*(- 
12*x^2 + 12*x^3) + E^9*(-1500*x^2 + 1500*x^3) + E^8*(-3*x^3 + E^3*(-180*x^ 
2 + 180*x^3)) + E^4*(-30*E^3*x^3 + E^6*(-900*x^2 + 900*x^3))) + E^x*(15*E^ 
3*x^3 + E^6*(600*x^2 - 600*x^3) + E^12*(48*x - 96*x^2 + 48*x^3) + E^9*(600 
0*x - 12000*x^2 + 6000*x^3) + E^8*(24*x^2 - 24*x^3 + E^3*(720*x - 1440*x^2 
 + 720*x^3)) + E^4*(3*x^3 + E^3*(240*x^2 - 240*x^3) + E^6*(3600*x - 7200*x 
^2 + 3600*x^3)))),x]
 

Output:

E^((2*E^6*(5 + E)^2*x^2)/(20*E^3*(-1 + x) + 4*E^4*(-1 + x) - x + 5*E^(3 + 
x)*x + E^(4 + x)*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.

\(\displaystyle \int \frac {\left (e^x \left (-4 e^{12} x^3-60 e^{11} x^3-300 e^{10} x^3-500 e^9 x^3\right )-16 e^{12} x-240 e^{11} x-1200 e^{10} x-2000 e^9 x\right ) \exp \left (\frac {2 e^8 x^2+20 e^7 x^2+50 e^6 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16 x^2-32 x+16\right )+e^6 \left (400 x^2-800 x+400\right )+e^{2 x} \left (e^8 x^2+10 e^7 x^2+25 e^6 x^2\right )+e^4 \left (-8 x^2+e^3 \left (160 x^2-320 x+160\right )+8 x\right )+e^x \left (-10 e^3 x^2+e^8 \left (8 x^2-8 x\right )+e^6 \left (200 x^2-200 x\right )+e^4 \left (e^3 \left (80 x^2-80 x\right )-2 x^2\right )\right )}\right )}{-x^3+e^{3 x} \left (e^{12} x^3+15 e^{11} x^3+75 e^{10} x^3+125 e^9 x^3\right )+e^6 \left (-1200 x^3+2400 x^2-1200 x\right )+e^3 \left (60 x^3-60 x^2\right )+e^{12} \left (64 x^3-192 x^2+192 x-64\right )+e^9 \left (8000 x^3-24000 x^2+24000 x-8000\right )+e^8 \left (-48 x^3+96 x^2+e^3 \left (960 x^3-2880 x^2+2880 x-960\right )-48 x\right )+e^4 \left (12 x^3-12 x^2+e^3 \left (-480 x^3+960 x^2-480 x\right )+e^6 \left (4800 x^3-14400 x^2+14400 x-4800\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (12 x^3-12 x^2\right )+e^9 \left (1500 x^3-1500 x^2\right )+e^8 \left (e^3 \left (180 x^3-180 x^2\right )-3 x^3\right )+e^4 \left (e^6 \left (900 x^3-900 x^2\right )-30 e^3 x^3\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x^3-96 x^2+48 x\right )+e^9 \left (6000 x^3-12000 x^2+6000 x\right )+e^8 \left (-24 x^3+24 x^2+e^3 \left (720 x^3-1440 x^2+720 x\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x^3-7200 x^2+3600 x\right )\right )\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (e^x \left (-4 e^{12} x^3-60 e^{11} x^3-300 e^{10} x^3-500 e^9 x^3\right )+\left (-2000 e^9-1200 e^{10}\right ) x-16 e^{12} x-240 e^{11} x\right ) \exp \left (\frac {2 e^8 x^2+20 e^7 x^2+50 e^6 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16 x^2-32 x+16\right )+e^6 \left (400 x^2-800 x+400\right )+e^{2 x} \left (e^8 x^2+10 e^7 x^2+25 e^6 x^2\right )+e^4 \left (-8 x^2+e^3 \left (160 x^2-320 x+160\right )+8 x\right )+e^x \left (-10 e^3 x^2+e^8 \left (8 x^2-8 x\right )+e^6 \left (200 x^2-200 x\right )+e^4 \left (e^3 \left (80 x^2-80 x\right )-2 x^2\right )\right )}\right )}{-x^3+e^{3 x} \left (e^{12} x^3+15 e^{11} x^3+75 e^{10} x^3+125 e^9 x^3\right )+e^6 \left (-1200 x^3+2400 x^2-1200 x\right )+e^3 \left (60 x^3-60 x^2\right )+e^{12} \left (64 x^3-192 x^2+192 x-64\right )+e^9 \left (8000 x^3-24000 x^2+24000 x-8000\right )+e^8 \left (-48 x^3+96 x^2+e^3 \left (960 x^3-2880 x^2+2880 x-960\right )-48 x\right )+e^4 \left (12 x^3-12 x^2+e^3 \left (-480 x^3+960 x^2-480 x\right )+e^6 \left (4800 x^3-14400 x^2+14400 x-4800\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (12 x^3-12 x^2\right )+e^9 \left (1500 x^3-1500 x^2\right )+e^8 \left (e^3 \left (180 x^3-180 x^2\right )-3 x^3\right )+e^4 \left (e^6 \left (900 x^3-900 x^2\right )-30 e^3 x^3\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x^3-96 x^2+48 x\right )+e^9 \left (6000 x^3-12000 x^2+6000 x\right )+e^8 \left (-24 x^3+24 x^2+e^3 \left (720 x^3-1440 x^2+720 x\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x^3-7200 x^2+3600 x\right )\right )\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (e^x \left (-4 e^{12} x^3-60 e^{11} x^3-300 e^{10} x^3-500 e^9 x^3\right )+\left (-240 e^{11}-16 e^{12}\right ) x+\left (-2000 e^9-1200 e^{10}\right ) x\right ) \exp \left (\frac {2 e^8 x^2+20 e^7 x^2+50 e^6 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16 x^2-32 x+16\right )+e^6 \left (400 x^2-800 x+400\right )+e^{2 x} \left (e^8 x^2+10 e^7 x^2+25 e^6 x^2\right )+e^4 \left (-8 x^2+e^3 \left (160 x^2-320 x+160\right )+8 x\right )+e^x \left (-10 e^3 x^2+e^8 \left (8 x^2-8 x\right )+e^6 \left (200 x^2-200 x\right )+e^4 \left (e^3 \left (80 x^2-80 x\right )-2 x^2\right )\right )}\right )}{-x^3+e^{3 x} \left (e^{12} x^3+15 e^{11} x^3+75 e^{10} x^3+125 e^9 x^3\right )+e^6 \left (-1200 x^3+2400 x^2-1200 x\right )+e^3 \left (60 x^3-60 x^2\right )+e^{12} \left (64 x^3-192 x^2+192 x-64\right )+e^9 \left (8000 x^3-24000 x^2+24000 x-8000\right )+e^8 \left (-48 x^3+96 x^2+e^3 \left (960 x^3-2880 x^2+2880 x-960\right )-48 x\right )+e^4 \left (12 x^3-12 x^2+e^3 \left (-480 x^3+960 x^2-480 x\right )+e^6 \left (4800 x^3-14400 x^2+14400 x-4800\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (12 x^3-12 x^2\right )+e^9 \left (1500 x^3-1500 x^2\right )+e^8 \left (e^3 \left (180 x^3-180 x^2\right )-3 x^3\right )+e^4 \left (e^6 \left (900 x^3-900 x^2\right )-30 e^3 x^3\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x^3-96 x^2+48 x\right )+e^9 \left (6000 x^3-12000 x^2+6000 x\right )+e^8 \left (-24 x^3+24 x^2+e^3 \left (720 x^3-1440 x^2+720 x\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x^3-7200 x^2+3600 x\right )\right )\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (e^x \left (-4 e^{12} x^3-60 e^{11} x^3-300 e^{10} x^3-500 e^9 x^3\right )+\left (-2000 e^9-1200 e^{10}-240 e^{11}-16 e^{12}\right ) x\right ) \exp \left (\frac {2 e^8 x^2+20 e^7 x^2+50 e^6 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16 x^2-32 x+16\right )+e^6 \left (400 x^2-800 x+400\right )+e^{2 x} \left (e^8 x^2+10 e^7 x^2+25 e^6 x^2\right )+e^4 \left (-8 x^2+e^3 \left (160 x^2-320 x+160\right )+8 x\right )+e^x \left (-10 e^3 x^2+e^8 \left (8 x^2-8 x\right )+e^6 \left (200 x^2-200 x\right )+e^4 \left (e^3 \left (80 x^2-80 x\right )-2 x^2\right )\right )}\right )}{-x^3+e^{3 x} \left (e^{12} x^3+15 e^{11} x^3+75 e^{10} x^3+125 e^9 x^3\right )+e^6 \left (-1200 x^3+2400 x^2-1200 x\right )+e^3 \left (60 x^3-60 x^2\right )+e^{12} \left (64 x^3-192 x^2+192 x-64\right )+e^9 \left (8000 x^3-24000 x^2+24000 x-8000\right )+e^8 \left (-48 x^3+96 x^2+e^3 \left (960 x^3-2880 x^2+2880 x-960\right )-48 x\right )+e^4 \left (12 x^3-12 x^2+e^3 \left (-480 x^3+960 x^2-480 x\right )+e^6 \left (4800 x^3-14400 x^2+14400 x-4800\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (12 x^3-12 x^2\right )+e^9 \left (1500 x^3-1500 x^2\right )+e^8 \left (e^3 \left (180 x^3-180 x^2\right )-3 x^3\right )+e^4 \left (e^6 \left (900 x^3-900 x^2\right )-30 e^3 x^3\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x^3-96 x^2+48 x\right )+e^9 \left (6000 x^3-12000 x^2+6000 x\right )+e^8 \left (-24 x^3+24 x^2+e^3 \left (720 x^3-1440 x^2+720 x\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x^3-7200 x^2+3600 x\right )\right )\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 (5+e)^3 x \left (-e^x x^2-4\right ) \exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 \left (1+\frac {1}{25} e (10+e)\right ) e^{2 x+6} x^2+9 x^2+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )+1800 \left (1+\frac {1}{25} e (10+e)\right ) e^{x+6} (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {9 e^{2 x+6} (5+e)^2 x^2-18 e^{x+3} (5+e) x^2+9 x^2-72 e^{x+6} (5+e)^2 (1-x) x+72 e^3 (5+e) (1-x) x+2 e^6 (5+e)^2 \left (73 x^2-144 x+72\right )}{\left (4 e^3 (5+e) (1-x)-e^{x+3} (5+e) x+x\right )^2}\right ) x \left (e^x x^2+4\right )}{\left (4 e^3 (5+e) (1-x)-e^{x+3} (5+e) x+x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {9 e^{2 x+6} (5+e)^2 x^2-18 e^{x+3} (5+e) x^2+9 x^2-72 e^{x+6} (5+e)^2 (1-x) x+72 e^3 (5+e) (1-x) x+2 e^6 (5+e)^2 \left (73 x^2-144 x+72\right )}{\left (4 e^3 (5+e) (1-x)-e^{x+3} (5+e) x+x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {9 e^{2 x+6} (5+e)^2 x^2-18 e^{x+3} (5+e) x^2+9 x^2-72 e^{x+6} (5+e)^2 (1-x) x+72 e^3 (5+e) (1-x) x+2 e^6 (5+e)^2 \left (73 x^2-144 x+72\right )}{\left (4 e^3 (5+e) (1-x)-e^{x+3} (5+e) x+x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 4 (5+e)^3 \int \frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}\right ) x \left (-e^x x^2-4\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 (5+e)^3 \int \left (\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) x^2}{(-5-e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^2}+\frac {\exp \left (\frac {-90 \left (1+\frac {e}{5}\right ) e^{x+3} x^2+225 e^{2 x+6} \left (1+\frac {1}{25} e (10+e)\right ) x^2+9 x^2+1800 e^{x+6} \left (1+\frac {1}{25} e (10+e)\right ) (x-1) x-360 \left (1+\frac {e}{5}\right ) e^3 (x-1) x+50 e^6 \left (1+\frac {1}{25} e (10+e)\right ) \left (73 x^2-144 x+72\right )}{\left (20 \left (1+\frac {e}{5}\right ) e^3 (x-1)+5 \left (1+\frac {e}{5}\right ) e^{x+3} x-x\right )^2}-3\right ) \left (\left (1-20 e^3-4 e^4\right ) x^2+4 e^3 (5+e) x+4 e^3 (5+e)\right ) x}{(5+e) \left (-5 \left (1+\frac {e}{5}\right ) e^{x+3} x+\left (1-4 e^3 (5+e)\right ) x+20 \left (1+\frac {e}{5}\right ) e^3\right )^3}\right )dx\)

Input:

Int[(E^((50*E^6*x^2 + 20*E^7*x^2 + 2*E^8*x^2)/(x^2 + E^3*(40*x - 40*x^2) + 
 E^8*(16 - 32*x + 16*x^2) + E^6*(400 - 800*x + 400*x^2) + E^(2*x)*(25*E^6* 
x^2 + 10*E^7*x^2 + E^8*x^2) + E^4*(8*x - 8*x^2 + E^3*(160 - 320*x + 160*x^ 
2)) + E^x*(-10*E^3*x^2 + E^8*(-8*x + 8*x^2) + E^6*(-200*x + 200*x^2) + E^4 
*(-2*x^2 + E^3*(-80*x + 80*x^2)))))*(-2000*E^9*x - 1200*E^10*x - 240*E^11* 
x - 16*E^12*x + E^x*(-500*E^9*x^3 - 300*E^10*x^3 - 60*E^11*x^3 - 4*E^12*x^ 
3)))/(-x^3 + E^6*(-1200*x + 2400*x^2 - 1200*x^3) + E^3*(-60*x^2 + 60*x^3) 
+ E^12*(-64 + 192*x - 192*x^2 + 64*x^3) + E^9*(-8000 + 24000*x - 24000*x^2 
 + 8000*x^3) + E^(3*x)*(125*E^9*x^3 + 75*E^10*x^3 + 15*E^11*x^3 + E^12*x^3 
) + E^8*(-48*x + 96*x^2 - 48*x^3 + E^3*(-960 + 2880*x - 2880*x^2 + 960*x^3 
)) + E^4*(-12*x^2 + 12*x^3 + E^3*(-480*x + 960*x^2 - 480*x^3) + E^6*(-4800 
 + 14400*x - 14400*x^2 + 4800*x^3)) + E^(2*x)*(-75*E^6*x^3 + E^12*(-12*x^2 
 + 12*x^3) + E^9*(-1500*x^2 + 1500*x^3) + E^8*(-3*x^3 + E^3*(-180*x^2 + 18 
0*x^3)) + E^4*(-30*E^3*x^3 + E^6*(-900*x^2 + 900*x^3))) + E^x*(15*E^3*x^3 
+ E^6*(600*x^2 - 600*x^3) + E^12*(48*x - 96*x^2 + 48*x^3) + E^9*(6000*x - 
12000*x^2 + 6000*x^3) + E^8*(24*x^2 - 24*x^3 + E^3*(720*x - 1440*x^2 + 720 
*x^3)) + E^4*(3*x^3 + E^3*(240*x^2 - 240*x^3) + E^6*(3600*x - 7200*x^2 + 3 
600*x^3)))),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(27)=54\).

Time = 104.78 (sec) , antiderivative size = 197, normalized size of antiderivative = 6.35

method	result	size
risch	\({\mathrm e}^{-\frac {2 x^{2} \left (10 \,{\mathrm e}^{7}+{\mathrm e}^{8}+25 \,{\mathrm e}^{6}\right )}{800 x \,{\mathrm e}^{6}-160 \,{\mathrm e}^{7} x^{2}+8 x \,{\mathrm e}^{x +8}+32 x \,{\mathrm e}^{8}+320 x \,{\mathrm e}^{7}-x^{2} {\mathrm e}^{2 x +8}-40 x \,{\mathrm e}^{3}+40 x^{2} {\mathrm e}^{3}-80 x^{2} {\mathrm e}^{x +7}+80 x \,{\mathrm e}^{x +7}-160 \,{\mathrm e}^{7}-16 \,{\mathrm e}^{8}-400 \,{\mathrm e}^{6}+8 x^{2} {\mathrm e}^{4}-8 x \,{\mathrm e}^{4}+10 x^{2} {\mathrm e}^{3+x}-x^{2}-400 x^{2} {\mathrm e}^{6}+2 x^{2} {\mathrm e}^{4+x}-8 x^{2} {\mathrm e}^{x +8}-25 x^{2} {\mathrm e}^{2 x +6}+200 x \,{\mathrm e}^{6+x}-200 x^{2} {\mathrm e}^{6+x}-10 x^{2} {\mathrm e}^{2 x +7}-16 \,{\mathrm e}^{8} x^{2}}}\)	\(197\)
parallelrisch	\(\text {Expression too large to display}\)	\(1762\)

Input:

int(((-4*x^3*exp(4)^3-60*x^3*exp(3)*exp(4)^2-300*x^3*exp(3)^2*exp(4)-500*x 
^3*exp(3)^3)*exp(x)-16*x*exp(4)^3-240*x*exp(3)*exp(4)^2-1200*x*exp(3)^2*ex 
p(4)-2000*x*exp(3)^3)*exp((2*x^2*exp(4)^2+20*x^2*exp(3)*exp(4)+50*x^2*exp( 
3)^2)/((x^2*exp(4)^2+10*x^2*exp(3)*exp(4)+25*x^2*exp(3)^2)*exp(x)^2+((8*x^ 
2-8*x)*exp(4)^2+((80*x^2-80*x)*exp(3)-2*x^2)*exp(4)+(200*x^2-200*x)*exp(3) 
^2-10*x^2*exp(3))*exp(x)+(16*x^2-32*x+16)*exp(4)^2+((160*x^2-320*x+160)*ex 
p(3)-8*x^2+8*x)*exp(4)+(400*x^2-800*x+400)*exp(3)^2+(-40*x^2+40*x)*exp(3)+ 
x^2))/((x^3*exp(4)^3+15*x^3*exp(3)*exp(4)^2+75*x^3*exp(3)^2*exp(4)+125*x^3 
*exp(3)^3)*exp(x)^3+((12*x^3-12*x^2)*exp(4)^3+((180*x^3-180*x^2)*exp(3)-3* 
x^3)*exp(4)^2+((900*x^3-900*x^2)*exp(3)^2-30*x^3*exp(3))*exp(4)+(1500*x^3- 
1500*x^2)*exp(3)^3-75*x^3*exp(3)^2)*exp(x)^2+((48*x^3-96*x^2+48*x)*exp(4)^ 
3+((720*x^3-1440*x^2+720*x)*exp(3)-24*x^3+24*x^2)*exp(4)^2+((3600*x^3-7200 
*x^2+3600*x)*exp(3)^2+(-240*x^3+240*x^2)*exp(3)+3*x^3)*exp(4)+(6000*x^3-12 
000*x^2+6000*x)*exp(3)^3+(-600*x^3+600*x^2)*exp(3)^2+15*x^3*exp(3))*exp(x) 
+(64*x^3-192*x^2+192*x-64)*exp(4)^3+((960*x^3-2880*x^2+2880*x-960)*exp(3)- 
48*x^3+96*x^2-48*x)*exp(4)^2+((4800*x^3-14400*x^2+14400*x-4800)*exp(3)^2+( 
-480*x^3+960*x^2-480*x)*exp(3)+12*x^3-12*x^2)*exp(4)+(8000*x^3-24000*x^2+2 
4000*x-8000)*exp(3)^3+(-1200*x^3+2400*x^2-1200*x)*exp(3)^2+(60*x^3-60*x^2) 
*exp(3)-x^3),x,method=_RETURNVERBOSE)
 

Output:

exp(-2*x^2*(10*exp(7)+exp(8)+25*exp(6))/(800*x*exp(6)-160*exp(7)*x^2+8*x*e 
xp(x+8)+32*x*exp(8)+320*x*exp(7)-x^2*exp(2*x+8)-40*x*exp(3)+40*x^2*exp(3)- 
80*x^2*exp(x+7)+80*x*exp(x+7)-160*exp(7)-16*exp(8)-400*exp(6)+8*x^2*exp(4) 
-8*x*exp(4)+10*x^2*exp(3+x)-x^2-400*x^2*exp(6)+2*x^2*exp(4+x)-8*x^2*exp(x+ 
8)-25*x^2*exp(2*x+6)+200*x*exp(6+x)-200*x^2*exp(6+x)-10*x^2*exp(2*x+7)-16* 
exp(8)*x^2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (28) = 56\).

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.32 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=e^{\left (\frac {2 \, {\left (x^{2} e^{8} + 10 \, x^{2} e^{7} + 25 \, x^{2} e^{6}\right )}}{x^{2} + 16 \, {\left (x^{2} - 2 \, x + 1\right )} e^{8} + 160 \, {\left (x^{2} - 2 \, x + 1\right )} e^{7} + 400 \, {\left (x^{2} - 2 \, x + 1\right )} e^{6} - 8 \, {\left (x^{2} - x\right )} e^{4} - 40 \, {\left (x^{2} - x\right )} e^{3} + {\left (x^{2} e^{8} + 10 \, x^{2} e^{7} + 25 \, x^{2} e^{6}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} e^{4} + 5 \, x^{2} e^{3} - 4 \, {\left (x^{2} - x\right )} e^{8} - 40 \, {\left (x^{2} - x\right )} e^{7} - 100 \, {\left (x^{2} - x\right )} e^{6}\right )} e^{x}}\right )} \] Input:

integrate(((-4*x^3*exp(4)^3-60*x^3*exp(3)*exp(4)^2-300*x^3*exp(3)^2*exp(4) 
-500*x^3*exp(3)^3)*exp(x)-16*x*exp(4)^3-240*x*exp(3)*exp(4)^2-1200*x*exp(3 
)^2*exp(4)-2000*x*exp(3)^3)*exp((2*x^2*exp(4)^2+20*x^2*exp(3)*exp(4)+50*x^ 
2*exp(3)^2)/((x^2*exp(4)^2+10*x^2*exp(3)*exp(4)+25*x^2*exp(3)^2)*exp(x)^2+ 
((8*x^2-8*x)*exp(4)^2+((80*x^2-80*x)*exp(3)-2*x^2)*exp(4)+(200*x^2-200*x)* 
exp(3)^2-10*x^2*exp(3))*exp(x)+(16*x^2-32*x+16)*exp(4)^2+((160*x^2-320*x+1 
60)*exp(3)-8*x^2+8*x)*exp(4)+(400*x^2-800*x+400)*exp(3)^2+(-40*x^2+40*x)*e 
xp(3)+x^2))/((x^3*exp(4)^3+15*x^3*exp(3)*exp(4)^2+75*x^3*exp(3)^2*exp(4)+1 
25*x^3*exp(3)^3)*exp(x)^3+((12*x^3-12*x^2)*exp(4)^3+((180*x^3-180*x^2)*exp 
(3)-3*x^3)*exp(4)^2+((900*x^3-900*x^2)*exp(3)^2-30*x^3*exp(3))*exp(4)+(150 
0*x^3-1500*x^2)*exp(3)^3-75*x^3*exp(3)^2)*exp(x)^2+((48*x^3-96*x^2+48*x)*e 
xp(4)^3+((720*x^3-1440*x^2+720*x)*exp(3)-24*x^3+24*x^2)*exp(4)^2+((3600*x^ 
3-7200*x^2+3600*x)*exp(3)^2+(-240*x^3+240*x^2)*exp(3)+3*x^3)*exp(4)+(6000* 
x^3-12000*x^2+6000*x)*exp(3)^3+(-600*x^3+600*x^2)*exp(3)^2+15*x^3*exp(3))* 
exp(x)+(64*x^3-192*x^2+192*x-64)*exp(4)^3+((960*x^3-2880*x^2+2880*x-960)*e 
xp(3)-48*x^3+96*x^2-48*x)*exp(4)^2+((4800*x^3-14400*x^2+14400*x-4800)*exp( 
3)^2+(-480*x^3+960*x^2-480*x)*exp(3)+12*x^3-12*x^2)*exp(4)+(8000*x^3-24000 
*x^2+24000*x-8000)*exp(3)^3+(-1200*x^3+2400*x^2-1200*x)*exp(3)^2+(60*x^3-6 
0*x^2)*exp(3)-x^3),x, algorithm="fricas")
 

Output:

e^(2*(x^2*e^8 + 10*x^2*e^7 + 25*x^2*e^6)/(x^2 + 16*(x^2 - 2*x + 1)*e^8 + 1 
60*(x^2 - 2*x + 1)*e^7 + 400*(x^2 - 2*x + 1)*e^6 - 8*(x^2 - x)*e^4 - 40*(x 
^2 - x)*e^3 + (x^2*e^8 + 10*x^2*e^7 + 25*x^2*e^6)*e^(2*x) - 2*(x^2*e^4 + 5 
*x^2*e^3 - 4*(x^2 - x)*e^8 - 40*(x^2 - x)*e^7 - 100*(x^2 - x)*e^6)*e^x))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (22) = 44\).

Time = 1.96 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.74 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=e^{\frac {2 x^{2} e^{8} + 50 x^{2} e^{6} + 20 x^{2} e^{7}}{x^{2} + \left (- 40 x^{2} + 40 x\right ) e^{3} + \left (- 8 x^{2} + 8 x + \left (160 x^{2} - 320 x + 160\right ) e^{3}\right ) e^{4} + \left (16 x^{2} - 32 x + 16\right ) e^{8} + \left (400 x^{2} - 800 x + 400\right ) e^{6} + \left (x^{2} e^{8} + 25 x^{2} e^{6} + 10 x^{2} e^{7}\right ) e^{2 x} + \left (- 10 x^{2} e^{3} + \left (- 2 x^{2} + \left (80 x^{2} - 80 x\right ) e^{3}\right ) e^{4} + \left (8 x^{2} - 8 x\right ) e^{8} + \left (200 x^{2} - 200 x\right ) e^{6}\right ) e^{x}}} \] Input:

integrate(((-4*x**3*exp(4)**3-60*x**3*exp(3)*exp(4)**2-300*x**3*exp(3)**2* 
exp(4)-500*x**3*exp(3)**3)*exp(x)-16*x*exp(4)**3-240*x*exp(3)*exp(4)**2-12 
00*x*exp(3)**2*exp(4)-2000*x*exp(3)**3)*exp((2*x**2*exp(4)**2+20*x**2*exp( 
3)*exp(4)+50*x**2*exp(3)**2)/((x**2*exp(4)**2+10*x**2*exp(3)*exp(4)+25*x** 
2*exp(3)**2)*exp(x)**2+((8*x**2-8*x)*exp(4)**2+((80*x**2-80*x)*exp(3)-2*x* 
*2)*exp(4)+(200*x**2-200*x)*exp(3)**2-10*x**2*exp(3))*exp(x)+(16*x**2-32*x 
+16)*exp(4)**2+((160*x**2-320*x+160)*exp(3)-8*x**2+8*x)*exp(4)+(400*x**2-8 
00*x+400)*exp(3)**2+(-40*x**2+40*x)*exp(3)+x**2))/((x**3*exp(4)**3+15*x**3 
*exp(3)*exp(4)**2+75*x**3*exp(3)**2*exp(4)+125*x**3*exp(3)**3)*exp(x)**3+( 
(12*x**3-12*x**2)*exp(4)**3+((180*x**3-180*x**2)*exp(3)-3*x**3)*exp(4)**2+ 
((900*x**3-900*x**2)*exp(3)**2-30*x**3*exp(3))*exp(4)+(1500*x**3-1500*x**2 
)*exp(3)**3-75*x**3*exp(3)**2)*exp(x)**2+((48*x**3-96*x**2+48*x)*exp(4)**3 
+((720*x**3-1440*x**2+720*x)*exp(3)-24*x**3+24*x**2)*exp(4)**2+((3600*x**3 
-7200*x**2+3600*x)*exp(3)**2+(-240*x**3+240*x**2)*exp(3)+3*x**3)*exp(4)+(6 
000*x**3-12000*x**2+6000*x)*exp(3)**3+(-600*x**3+600*x**2)*exp(3)**2+15*x* 
*3*exp(3))*exp(x)+(64*x**3-192*x**2+192*x-64)*exp(4)**3+((960*x**3-2880*x* 
*2+2880*x-960)*exp(3)-48*x**3+96*x**2-48*x)*exp(4)**2+((4800*x**3-14400*x* 
*2+14400*x-4800)*exp(3)**2+(-480*x**3+960*x**2-480*x)*exp(3)+12*x**3-12*x* 
*2)*exp(4)+(8000*x**3-24000*x**2+24000*x-8000)*exp(3)**3+(-1200*x**3+2400* 
x**2-1200*x)*exp(3)**2+(60*x**3-60*x**2)*exp(3)-x**3),x)
 

Output:

exp((2*x**2*exp(8) + 50*x**2*exp(6) + 20*x**2*exp(7))/(x**2 + (-40*x**2 + 
40*x)*exp(3) + (-8*x**2 + 8*x + (160*x**2 - 320*x + 160)*exp(3))*exp(4) + 
(16*x**2 - 32*x + 16)*exp(8) + (400*x**2 - 800*x + 400)*exp(6) + (x**2*exp 
(8) + 25*x**2*exp(6) + 10*x**2*exp(7))*exp(2*x) + (-10*x**2*exp(3) + (-2*x 
**2 + (80*x**2 - 80*x)*exp(3))*exp(4) + (8*x**2 - 8*x)*exp(8) + (200*x**2 
- 200*x)*exp(6))*exp(x)))
 

Maxima [F(-1)]

Timed out. \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((-4*x^3*exp(4)^3-60*x^3*exp(3)*exp(4)^2-300*x^3*exp(3)^2*exp(4) 
-500*x^3*exp(3)^3)*exp(x)-16*x*exp(4)^3-240*x*exp(3)*exp(4)^2-1200*x*exp(3 
)^2*exp(4)-2000*x*exp(3)^3)*exp((2*x^2*exp(4)^2+20*x^2*exp(3)*exp(4)+50*x^ 
2*exp(3)^2)/((x^2*exp(4)^2+10*x^2*exp(3)*exp(4)+25*x^2*exp(3)^2)*exp(x)^2+ 
((8*x^2-8*x)*exp(4)^2+((80*x^2-80*x)*exp(3)-2*x^2)*exp(4)+(200*x^2-200*x)* 
exp(3)^2-10*x^2*exp(3))*exp(x)+(16*x^2-32*x+16)*exp(4)^2+((160*x^2-320*x+1 
60)*exp(3)-8*x^2+8*x)*exp(4)+(400*x^2-800*x+400)*exp(3)^2+(-40*x^2+40*x)*e 
xp(3)+x^2))/((x^3*exp(4)^3+15*x^3*exp(3)*exp(4)^2+75*x^3*exp(3)^2*exp(4)+1 
25*x^3*exp(3)^3)*exp(x)^3+((12*x^3-12*x^2)*exp(4)^3+((180*x^3-180*x^2)*exp 
(3)-3*x^3)*exp(4)^2+((900*x^3-900*x^2)*exp(3)^2-30*x^3*exp(3))*exp(4)+(150 
0*x^3-1500*x^2)*exp(3)^3-75*x^3*exp(3)^2)*exp(x)^2+((48*x^3-96*x^2+48*x)*e 
xp(4)^3+((720*x^3-1440*x^2+720*x)*exp(3)-24*x^3+24*x^2)*exp(4)^2+((3600*x^ 
3-7200*x^2+3600*x)*exp(3)^2+(-240*x^3+240*x^2)*exp(3)+3*x^3)*exp(4)+(6000* 
x^3-12000*x^2+6000*x)*exp(3)^3+(-600*x^3+600*x^2)*exp(3)^2+15*x^3*exp(3))* 
exp(x)+(64*x^3-192*x^2+192*x-64)*exp(4)^3+((960*x^3-2880*x^2+2880*x-960)*e 
xp(3)-48*x^3+96*x^2-48*x)*exp(4)^2+((4800*x^3-14400*x^2+14400*x-4800)*exp( 
3)^2+(-480*x^3+960*x^2-480*x)*exp(3)+12*x^3-12*x^2)*exp(4)+(8000*x^3-24000 
*x^2+24000*x-8000)*exp(3)^3+(-1200*x^3+2400*x^2-1200*x)*exp(3)^2+(60*x^3-6 
0*x^2)*exp(3)-x^3),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((-4*x^3*exp(4)^3-60*x^3*exp(3)*exp(4)^2-300*x^3*exp(3)^2*exp(4) 
-500*x^3*exp(3)^3)*exp(x)-16*x*exp(4)^3-240*x*exp(3)*exp(4)^2-1200*x*exp(3 
)^2*exp(4)-2000*x*exp(3)^3)*exp((2*x^2*exp(4)^2+20*x^2*exp(3)*exp(4)+50*x^ 
2*exp(3)^2)/((x^2*exp(4)^2+10*x^2*exp(3)*exp(4)+25*x^2*exp(3)^2)*exp(x)^2+ 
((8*x^2-8*x)*exp(4)^2+((80*x^2-80*x)*exp(3)-2*x^2)*exp(4)+(200*x^2-200*x)* 
exp(3)^2-10*x^2*exp(3))*exp(x)+(16*x^2-32*x+16)*exp(4)^2+((160*x^2-320*x+1 
60)*exp(3)-8*x^2+8*x)*exp(4)+(400*x^2-800*x+400)*exp(3)^2+(-40*x^2+40*x)*e 
xp(3)+x^2))/((x^3*exp(4)^3+15*x^3*exp(3)*exp(4)^2+75*x^3*exp(3)^2*exp(4)+1 
25*x^3*exp(3)^3)*exp(x)^3+((12*x^3-12*x^2)*exp(4)^3+((180*x^3-180*x^2)*exp 
(3)-3*x^3)*exp(4)^2+((900*x^3-900*x^2)*exp(3)^2-30*x^3*exp(3))*exp(4)+(150 
0*x^3-1500*x^2)*exp(3)^3-75*x^3*exp(3)^2)*exp(x)^2+((48*x^3-96*x^2+48*x)*e 
xp(4)^3+((720*x^3-1440*x^2+720*x)*exp(3)-24*x^3+24*x^2)*exp(4)^2+((3600*x^ 
3-7200*x^2+3600*x)*exp(3)^2+(-240*x^3+240*x^2)*exp(3)+3*x^3)*exp(4)+(6000* 
x^3-12000*x^2+6000*x)*exp(3)^3+(-600*x^3+600*x^2)*exp(3)^2+15*x^3*exp(3))* 
exp(x)+(64*x^3-192*x^2+192*x-64)*exp(4)^3+((960*x^3-2880*x^2+2880*x-960)*e 
xp(3)-48*x^3+96*x^2-48*x)*exp(4)^2+((4800*x^3-14400*x^2+14400*x-4800)*exp( 
3)^2+(-480*x^3+960*x^2-480*x)*exp(3)+12*x^3-12*x^2)*exp(4)+(8000*x^3-24000 
*x^2+24000*x-8000)*exp(3)^3+(-1200*x^3+2400*x^2-1200*x)*exp(3)^2+(60*x^3-6 
0*x^2)*exp(3)-x^3),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 4.42 (sec) , antiderivative size = 553, normalized size of antiderivative = 17.84 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx =\text {Too large to display} \] Input:

int((exp((50*x^2*exp(6) + 20*x^2*exp(7) + 2*x^2*exp(8))/(exp(8)*(16*x^2 - 
32*x + 16) - exp(x)*(exp(8)*(8*x - 8*x^2) + exp(6)*(200*x - 200*x^2) + exp 
(4)*(exp(3)*(80*x - 80*x^2) + 2*x^2) + 10*x^2*exp(3)) + exp(3)*(40*x - 40* 
x^2) + exp(6)*(400*x^2 - 800*x + 400) + exp(4)*(8*x + exp(3)*(160*x^2 - 32 
0*x + 160) - 8*x^2) + exp(2*x)*(25*x^2*exp(6) + 10*x^2*exp(7) + x^2*exp(8) 
) + x^2))*(exp(x)*(500*x^3*exp(9) + 300*x^3*exp(10) + 60*x^3*exp(11) + 4*x 
^3*exp(12)) + 2000*x*exp(9) + 1200*x*exp(10) + 240*x*exp(11) + 16*x*exp(12 
)))/(exp(4)*(exp(3)*(480*x - 960*x^2 + 480*x^3) - exp(6)*(14400*x - 14400* 
x^2 + 4800*x^3 - 4800) + 12*x^2 - 12*x^3) - exp(12)*(192*x - 192*x^2 + 64* 
x^3 - 64) + exp(6)*(1200*x - 2400*x^2 + 1200*x^3) - exp(9)*(24000*x - 2400 
0*x^2 + 8000*x^3 - 8000) - exp(3*x)*(125*x^3*exp(9) + 75*x^3*exp(10) + 15* 
x^3*exp(11) + x^3*exp(12)) + exp(3)*(60*x^2 - 60*x^3) + exp(8)*(48*x - exp 
(3)*(2880*x - 2880*x^2 + 960*x^3 - 960) - 96*x^2 + 48*x^3) + x^3 - exp(x)* 
(exp(4)*(exp(6)*(3600*x - 7200*x^2 + 3600*x^3) + exp(3)*(240*x^2 - 240*x^3 
) + 3*x^3) + exp(12)*(48*x - 96*x^2 + 48*x^3) + exp(9)*(6000*x - 12000*x^2 
 + 6000*x^3) + exp(6)*(600*x^2 - 600*x^3) + 15*x^3*exp(3) + exp(8)*(exp(3) 
*(720*x - 1440*x^2 + 720*x^3) + 24*x^2 - 24*x^3)) + exp(2*x)*(exp(4)*(exp( 
6)*(900*x^2 - 900*x^3) + 30*x^3*exp(3)) + exp(12)*(12*x^2 - 12*x^3) + exp( 
9)*(1500*x^2 - 1500*x^3) + 75*x^3*exp(6) + exp(8)*(exp(3)*(180*x^2 - 180*x 
^3) + 3*x^3))),x)
 

Output:

exp((2*x^2*exp(8))/(400*exp(6) + 160*exp(7) + 16*exp(8) + 40*x*exp(3) + 8* 
x*exp(4) - 800*x*exp(6) - 320*x*exp(7) - 32*x*exp(8) - 40*x^2*exp(3) - 8*x 
^2*exp(4) + 400*x^2*exp(6) + 160*x^2*exp(7) + 16*x^2*exp(8) + x^2 - 200*x* 
exp(6)*exp(x) - 80*x*exp(7)*exp(x) - 8*x*exp(8)*exp(x) - 10*x^2*exp(3)*exp 
(x) - 2*x^2*exp(4)*exp(x) + 200*x^2*exp(6)*exp(x) + 80*x^2*exp(7)*exp(x) + 
 8*x^2*exp(8)*exp(x) + 25*x^2*exp(2*x)*exp(6) + 10*x^2*exp(2*x)*exp(7) + x 
^2*exp(2*x)*exp(8)))*exp((20*x^2*exp(7))/(400*exp(6) + 160*exp(7) + 16*exp 
(8) + 40*x*exp(3) + 8*x*exp(4) - 800*x*exp(6) - 320*x*exp(7) - 32*x*exp(8) 
 - 40*x^2*exp(3) - 8*x^2*exp(4) + 400*x^2*exp(6) + 160*x^2*exp(7) + 16*x^2 
*exp(8) + x^2 - 200*x*exp(6)*exp(x) - 80*x*exp(7)*exp(x) - 8*x*exp(8)*exp( 
x) - 10*x^2*exp(3)*exp(x) - 2*x^2*exp(4)*exp(x) + 200*x^2*exp(6)*exp(x) + 
80*x^2*exp(7)*exp(x) + 8*x^2*exp(8)*exp(x) + 25*x^2*exp(2*x)*exp(6) + 10*x 
^2*exp(2*x)*exp(7) + x^2*exp(2*x)*exp(8)))*exp((50*x^2*exp(6))/(400*exp(6) 
 + 160*exp(7) + 16*exp(8) + 40*x*exp(3) + 8*x*exp(4) - 800*x*exp(6) - 320* 
x*exp(7) - 32*x*exp(8) - 40*x^2*exp(3) - 8*x^2*exp(4) + 400*x^2*exp(6) + 1 
60*x^2*exp(7) + 16*x^2*exp(8) + x^2 - 200*x*exp(6)*exp(x) - 80*x*exp(7)*ex 
p(x) - 8*x*exp(8)*exp(x) - 10*x^2*exp(3)*exp(x) - 2*x^2*exp(4)*exp(x) + 20 
0*x^2*exp(6)*exp(x) + 80*x^2*exp(7)*exp(x) + 8*x^2*exp(8)*exp(x) + 25*x^2* 
exp(2*x)*exp(6) + 10*x^2*exp(2*x)*exp(7) + x^2*exp(2*x)*exp(8)))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 239, normalized size of antiderivative = 7.71 \[ \int \frac {e^{\frac {50 e^6 x^2+20 e^7 x^2+2 e^8 x^2}{x^2+e^3 \left (40 x-40 x^2\right )+e^8 \left (16-32 x+16 x^2\right )+e^6 \left (400-800 x+400 x^2\right )+e^{2 x} \left (25 e^6 x^2+10 e^7 x^2+e^8 x^2\right )+e^4 \left (8 x-8 x^2+e^3 \left (160-320 x+160 x^2\right )\right )+e^x \left (-10 e^3 x^2+e^8 \left (-8 x+8 x^2\right )+e^6 \left (-200 x+200 x^2\right )+e^4 \left (-2 x^2+e^3 \left (-80 x+80 x^2\right )\right )\right )}} \left (-2000 e^9 x-1200 e^{10} x-240 e^{11} x-16 e^{12} x+e^x \left (-500 e^9 x^3-300 e^{10} x^3-60 e^{11} x^3-4 e^{12} x^3\right )\right )}{-x^3+e^6 \left (-1200 x+2400 x^2-1200 x^3\right )+e^3 \left (-60 x^2+60 x^3\right )+e^{12} \left (-64+192 x-192 x^2+64 x^3\right )+e^9 \left (-8000+24000 x-24000 x^2+8000 x^3\right )+e^{3 x} \left (125 e^9 x^3+75 e^{10} x^3+15 e^{11} x^3+e^{12} x^3\right )+e^8 \left (-48 x+96 x^2-48 x^3+e^3 \left (-960+2880 x-2880 x^2+960 x^3\right )\right )+e^4 \left (-12 x^2+12 x^3+e^3 \left (-480 x+960 x^2-480 x^3\right )+e^6 \left (-4800+14400 x-14400 x^2+4800 x^3\right )\right )+e^{2 x} \left (-75 e^6 x^3+e^{12} \left (-12 x^2+12 x^3\right )+e^9 \left (-1500 x^2+1500 x^3\right )+e^8 \left (-3 x^3+e^3 \left (-180 x^2+180 x^3\right )\right )+e^4 \left (-30 e^3 x^3+e^6 \left (-900 x^2+900 x^3\right )\right )\right )+e^x \left (15 e^3 x^3+e^6 \left (600 x^2-600 x^3\right )+e^{12} \left (48 x-96 x^2+48 x^3\right )+e^9 \left (6000 x-12000 x^2+6000 x^3\right )+e^8 \left (24 x^2-24 x^3+e^3 \left (720 x-1440 x^2+720 x^3\right )\right )+e^4 \left (3 x^3+e^3 \left (240 x^2-240 x^3\right )+e^6 \left (3600 x-7200 x^2+3600 x^3\right )\right )\right )} \, dx=e^{\frac {2 e^{8} x^{2}+20 e^{7} x^{2}+50 e^{6} x^{2}}{-8 e^{4} x^{2}-2 e^{x} e^{4} x^{2}-800 e^{6} x +16 e^{8} x^{2}+x^{2}-40 e^{3} x^{2}+40 e^{3} x +160 e^{7} x^{2}-320 e^{7} x +400 e^{6} x^{2}+8 e^{4} x +160 e^{7}+400 e^{6}+e^{2 x} e^{8} x^{2}-200 e^{x} e^{6} x +25 e^{2 x} e^{6} x^{2}-32 e^{8} x +16 e^{8}+10 e^{2 x} e^{7} x^{2}+8 e^{x} e^{8} x^{2}-8 e^{x} e^{8} x +80 e^{x} e^{7} x^{2}-80 e^{x} e^{7} x +200 e^{x} e^{6} x^{2}-10 e^{x} e^{3} x^{2}}} \] Input:

int(((-4*x^3*exp(4)^3-60*x^3*exp(3)*exp(4)^2-300*x^3*exp(3)^2*exp(4)-500*x 
^3*exp(3)^3)*exp(x)-16*x*exp(4)^3-240*x*exp(3)*exp(4)^2-1200*x*exp(3)^2*ex 
p(4)-2000*x*exp(3)^3)*exp((2*x^2*exp(4)^2+20*x^2*exp(3)*exp(4)+50*x^2*exp( 
3)^2)/((x^2*exp(4)^2+10*x^2*exp(3)*exp(4)+25*x^2*exp(3)^2)*exp(x)^2+((8*x^ 
2-8*x)*exp(4)^2+((80*x^2-80*x)*exp(3)-2*x^2)*exp(4)+(200*x^2-200*x)*exp(3) 
^2-10*x^2*exp(3))*exp(x)+(16*x^2-32*x+16)*exp(4)^2+((160*x^2-320*x+160)*ex 
p(3)-8*x^2+8*x)*exp(4)+(400*x^2-800*x+400)*exp(3)^2+(-40*x^2+40*x)*exp(3)+ 
x^2))/((x^3*exp(4)^3+15*x^3*exp(3)*exp(4)^2+75*x^3*exp(3)^2*exp(4)+125*x^3 
*exp(3)^3)*exp(x)^3+((12*x^3-12*x^2)*exp(4)^3+((180*x^3-180*x^2)*exp(3)-3* 
x^3)*exp(4)^2+((900*x^3-900*x^2)*exp(3)^2-30*x^3*exp(3))*exp(4)+(1500*x^3- 
1500*x^2)*exp(3)^3-75*x^3*exp(3)^2)*exp(x)^2+((48*x^3-96*x^2+48*x)*exp(4)^ 
3+((720*x^3-1440*x^2+720*x)*exp(3)-24*x^3+24*x^2)*exp(4)^2+((3600*x^3-7200 
*x^2+3600*x)*exp(3)^2+(-240*x^3+240*x^2)*exp(3)+3*x^3)*exp(4)+(6000*x^3-12 
000*x^2+6000*x)*exp(3)^3+(-600*x^3+600*x^2)*exp(3)^2+15*x^3*exp(3))*exp(x) 
+(64*x^3-192*x^2+192*x-64)*exp(4)^3+((960*x^3-2880*x^2+2880*x-960)*exp(3)- 
48*x^3+96*x^2-48*x)*exp(4)^2+((4800*x^3-14400*x^2+14400*x-4800)*exp(3)^2+( 
-480*x^3+960*x^2-480*x)*exp(3)+12*x^3-12*x^2)*exp(4)+(8000*x^3-24000*x^2+2 
4000*x-8000)*exp(3)^3+(-1200*x^3+2400*x^2-1200*x)*exp(3)^2+(60*x^3-60*x^2) 
*exp(3)-x^3),x)
 

Output:

e**((2*e**8*x**2 + 20*e**7*x**2 + 50*e**6*x**2)/(e**(2*x)*e**8*x**2 + 10*e 
**(2*x)*e**7*x**2 + 25*e**(2*x)*e**6*x**2 + 8*e**x*e**8*x**2 - 8*e**x*e**8 
*x + 80*e**x*e**7*x**2 - 80*e**x*e**7*x + 200*e**x*e**6*x**2 - 200*e**x*e* 
*6*x - 2*e**x*e**4*x**2 - 10*e**x*e**3*x**2 + 16*e**8*x**2 - 32*e**8*x + 1 
6*e**8 + 160*e**7*x**2 - 320*e**7*x + 160*e**7 + 400*e**6*x**2 - 800*e**6* 
x + 400*e**6 - 8*e**4*x**2 + 8*e**4*x - 40*e**3*x**2 + 40*e**3*x + x**2))