Integrand size = 333, antiderivative size = 26 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-79-x+\frac {x}{5+\left (e^x-x\right )^2 (-5+x)^2} \]
Time = 10.15 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=x \left (-1+\frac {1}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4}\right ) \]
Integrate[(-20 - 275*x^2 + 120*x^3 - 638*x^4 + 500*x^5 - 150*x^6 + 20*x^7 - x^8 + E^(4*x)*(-625 + 500*x - 150*x^2 + 20*x^3 - x^4) + E^(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) + E^(2*x)*(-225 + 50*x - 3741*x^2 + 2998*x^3 - 900*x^4 + 120*x^5 - 6*x^6) + E^x*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7))/(25 + 250*x^2 - 100*x^3 + 635*x^4 - 500*x^5 + 150*x^6 - 20*x^7 + x^8 + E^(4*x)*(625 - 500*x + 150*x^2 - 20*x^ 3 + x^4) + E^(3*x)*(-2500*x + 2000*x^2 - 600*x^3 + 80*x^4 - 4*x^5) + E^(2* x)*(250 - 100*x + 3760*x^2 - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6) + E^x*( -500*x + 200*x^2 - 2520*x^3 + 2000*x^4 - 600*x^5 + 80*x^6 - 4*x^7)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^8+20 x^7-150 x^6+500 x^5-638 x^4+120 x^3-275 x^2+e^{4 x} \left (-x^4+20 x^3-150 x^2+500 x-625\right )+e^{3 x} \left (4 x^5-80 x^4+600 x^3-2000 x^2+2500 x\right )+e^{2 x} \left (-6 x^6+120 x^5-900 x^4+2998 x^3-3741 x^2+50 x-225\right )+e^x \left (4 x^7-80 x^6+600 x^5-1998 x^4+2504 x^3-170 x^2+500 x\right )-20}{x^8-20 x^7+150 x^6-500 x^5+635 x^4-100 x^3+250 x^2+e^{4 x} \left (x^4-20 x^3+150 x^2-500 x+625\right )+e^{3 x} \left (-4 x^5+80 x^4-600 x^3+2000 x^2-2500 x\right )+e^{2 x} \left (6 x^6-120 x^5+900 x^4-3000 x^3+3760 x^2-100 x+250\right )+e^x \left (-4 x^7+80 x^6-600 x^5+2000 x^4-2520 x^3+200 x^2-500 x\right )+25} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-x^8+20 x^7-150 x^6+500 x^5-638 x^4+120 x^3-275 x^2+2 e^x \left (2 x^6-40 x^5+300 x^4-999 x^3+1252 x^2-85 x+250\right ) x+e^{2 x} \left (-6 x^6+120 x^5-900 x^4+2998 x^3-3741 x^2+50 x-225\right )+4 e^{3 x} (x-5)^4 x-e^{4 x} (x-5)^4-20}{\left (x^4-10 x^3+25 x^2-2 e^x (x-5)^2 x+e^{2 x} (x-5)^2+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^2-9 x+5}{(x-5) \left (x^4-2 e^x x^3-10 x^3+20 e^x x^2+e^{2 x} x^2+25 x^2-50 e^x x-10 e^{2 x} x+25 e^{2 x}+5\right )}+\frac {2 x \left (x^5-e^x x^4-16 x^4+16 e^x x^3+90 x^3-90 e^x x^2-200 x^2+200 e^x x+130 x-125 e^x-20\right )}{(x-5) \left (x^4-2 e^x x^3-10 x^3+20 e^x x^2+e^{2 x} x^2+25 x^2-50 e^x x-10 e^{2 x} x+25 e^{2 x}+5\right )^2}-1\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {2 x^2-9 x+5}{(x-5) \left (x^4-2 e^x x^3-10 x^3+20 e^x x^2+e^{2 x} x^2+25 x^2-50 e^x x-10 e^{2 x} x+25 e^{2 x}+5\right )}+\frac {2 x \left (x^5-e^x x^4-16 x^4+16 e^x x^3+90 x^3-90 e^x x^2-200 x^2+200 e^x x+130 x-125 e^x-20\right )}{(x-5) \left (x^4-2 e^x x^3-10 x^3+20 e^x x^2+e^{2 x} x^2+25 x^2-50 e^x x-10 e^{2 x} x+25 e^{2 x}+5\right )^2}-1\right )dx\) |
Int[(-20 - 275*x^2 + 120*x^3 - 638*x^4 + 500*x^5 - 150*x^6 + 20*x^7 - x^8 + E^(4*x)*(-625 + 500*x - 150*x^2 + 20*x^3 - x^4) + E^(3*x)*(2500*x - 2000 *x^2 + 600*x^3 - 80*x^4 + 4*x^5) + E^(2*x)*(-225 + 50*x - 3741*x^2 + 2998* x^3 - 900*x^4 + 120*x^5 - 6*x^6) + E^x*(500*x - 170*x^2 + 2504*x^3 - 1998* x^4 + 600*x^5 - 80*x^6 + 4*x^7))/(25 + 250*x^2 - 100*x^3 + 635*x^4 - 500*x ^5 + 150*x^6 - 20*x^7 + x^8 + E^(4*x)*(625 - 500*x + 150*x^2 - 20*x^3 + x^ 4) + E^(3*x)*(-2500*x + 2000*x^2 - 600*x^3 + 80*x^4 - 4*x^5) + E^(2*x)*(25 0 - 100*x + 3760*x^2 - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6) + E^x*(-500*x + 200*x^2 - 2520*x^3 + 2000*x^4 - 600*x^5 + 80*x^6 - 4*x^7)),x]
3.3.60.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(25)=50\).
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46
| method | result | size |
| risch | \(-x +\frac {x}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\) | \(64\) |
| parallelrisch | \(-\frac {x^{5}-2 \,{\mathrm e}^{x} x^{4}+50+{\mathrm e}^{2 x} x^{3}-75 x^{3}+150 \,{\mathrm e}^{x} x^{2}-75 x \,{\mathrm e}^{2 x}+250 x^{2}-500 \,{\mathrm e}^{x} x +250 \,{\mathrm e}^{2 x}+4 x}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\) | \(118\) |
| norman | \(\frac {-250 x^{2}-250 \,{\mathrm e}^{2 x}+75 x^{3}-150 \,{\mathrm e}^{x} x^{2}+75 x \,{\mathrm e}^{2 x}+500 \,{\mathrm e}^{x} x -4 x -x^{5}+2 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{3}-50}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\) | \(120\) |
int(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3-2000*x ^2+2500*x)*exp(x)^3+(-6*x^6+120*x^5-900*x^4+2998*x^3-3741*x^2+50*x-225)*ex p(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(x)-x^8+2 0*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x^2-500 *x+625)*exp(x)^4+(-4*x^5+80*x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6*x^6-1 20*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*x^6-600*x^ 5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5+635*x ^4-100*x^3+250*x^2+25),x,method=_RETURNVERBOSE)
-x+x/(exp(x)^2*x^2-2*exp(x)*x^3+x^4-10*x*exp(x)^2+20*exp(x)*x^2-10*x^3+25* exp(x)^2-50*exp(x)*x+25*x^2+5)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 10 \, x^{4} + 25 \, x^{3} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )} e^{x} + 4 \, x}{x^{4} - 10 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{x} + 5} \]
integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3- 2000*x^2+2500*x)*exp(x)^3+(-6*x^6+120*x^5-900*x^4+2998*x^3-3741*x^2+50*x-2 25)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(x) -x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x ^2-500*x+625)*exp(x)^4+(-4*x^5+80*x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6 *x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*x^6- 600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5 +635*x^4-100*x^3+250*x^2+25),x, algorithm=\
-(x^5 - 10*x^4 + 25*x^3 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 2*(x^4 - 10*x^3 + 25*x^2)*e^x + 4*x)/(x^4 - 10*x^3 + 25*x^2 + (x^2 - 10*x + 25)*e^(2*x) - 2*(x^3 - 10*x^2 + 25*x)*e^x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (17) = 34\).
Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=- x + \frac {x}{x^{4} - 10 x^{3} + 25 x^{2} + \left (x^{2} - 10 x + 25\right ) e^{2 x} + \left (- 2 x^{3} + 20 x^{2} - 50 x\right ) e^{x} + 5} \]
integrate(((-x**4+20*x**3-150*x**2+500*x-625)*exp(x)**4+(4*x**5-80*x**4+60 0*x**3-2000*x**2+2500*x)*exp(x)**3+(-6*x**6+120*x**5-900*x**4+2998*x**3-37 41*x**2+50*x-225)*exp(x)**2+(4*x**7-80*x**6+600*x**5-1998*x**4+2504*x**3-1 70*x**2+500*x)*exp(x)-x**8+20*x**7-150*x**6+500*x**5-638*x**4+120*x**3-275 *x**2-20)/((x**4-20*x**3+150*x**2-500*x+625)*exp(x)**4+(-4*x**5+80*x**4-60 0*x**3+2000*x**2-2500*x)*exp(x)**3+(6*x**6-120*x**5+900*x**4-3000*x**3+376 0*x**2-100*x+250)*exp(x)**2+(-4*x**7+80*x**6-600*x**5+2000*x**4-2520*x**3+ 200*x**2-500*x)*exp(x)+x**8-20*x**7+150*x**6-500*x**5+635*x**4-100*x**3+25 0*x**2+25),x)
-x + x/(x**4 - 10*x**3 + 25*x**2 + (x**2 - 10*x + 25)*exp(2*x) + (-2*x**3 + 20*x**2 - 50*x)*exp(x) + 5)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (25) = 50\).
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 10 \, x^{4} + 25 \, x^{3} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )} e^{x} + 4 \, x}{x^{4} - 10 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{x} + 5} \]
integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3- 2000*x^2+2500*x)*exp(x)^3+(-6*x^6+120*x^5-900*x^4+2998*x^3-3741*x^2+50*x-2 25)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(x) -x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x ^2-500*x+625)*exp(x)^4+(-4*x^5+80*x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6 *x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*x^6- 600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5 +635*x^4-100*x^3+250*x^2+25),x, algorithm=\
-(x^5 - 10*x^4 + 25*x^3 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 2*(x^4 - 10*x^3 + 25*x^2)*e^x + 4*x)/(x^4 - 10*x^3 + 25*x^2 + (x^2 - 10*x + 25)*e^(2*x) - 2*(x^3 - 10*x^2 + 25*x)*e^x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (25) = 50\).
Time = 0.56 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.65 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 2 \, x^{4} e^{x} - 10 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 20 \, x^{3} e^{x} + 25 \, x^{3} - 10 \, x^{2} e^{\left (2 \, x\right )} - 50 \, x^{2} e^{x} + 25 \, x e^{\left (2 \, x\right )} + 3 \, x}{x^{4} - 2 \, x^{3} e^{x} - 10 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 20 \, x^{2} e^{x} + 25 \, x^{2} - 10 \, x e^{\left (2 \, x\right )} - 50 \, x e^{x} + 25 \, e^{\left (2 \, x\right )} + 5} \]
integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3- 2000*x^2+2500*x)*exp(x)^3+(-6*x^6+120*x^5-900*x^4+2998*x^3-3741*x^2+50*x-2 25)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(x) -x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x ^2-500*x+625)*exp(x)^4+(-4*x^5+80*x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6 *x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*x^6- 600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5 +635*x^4-100*x^3+250*x^2+25),x, algorithm=\
-(x^5 - 2*x^4*e^x - 10*x^4 + x^3*e^(2*x) + 20*x^3*e^x + 25*x^3 - 10*x^2*e^ (2*x) - 50*x^2*e^x + 25*x*e^(2*x) + 3*x)/(x^4 - 2*x^3*e^x - 10*x^3 + x^2*e ^(2*x) + 20*x^2*e^x + 25*x^2 - 10*x*e^(2*x) - 50*x*e^x + 25*e^(2*x) + 5)
Timed out. \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=\int -\frac {{\mathrm {e}}^{4\,x}\,\left (x^4-20\,x^3+150\,x^2-500\,x+625\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x^5-80\,x^4+600\,x^3-2000\,x^2+2500\,x\right )-{\mathrm {e}}^x\,\left (4\,x^7-80\,x^6+600\,x^5-1998\,x^4+2504\,x^3-170\,x^2+500\,x\right )+275\,x^2-120\,x^3+638\,x^4-500\,x^5+150\,x^6-20\,x^7+x^8+{\mathrm {e}}^{2\,x}\,\left (6\,x^6-120\,x^5+900\,x^4-2998\,x^3+3741\,x^2-50\,x+225\right )+20}{{\mathrm {e}}^{4\,x}\,\left (x^4-20\,x^3+150\,x^2-500\,x+625\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x^5-80\,x^4+600\,x^3-2000\,x^2+2500\,x\right )-{\mathrm {e}}^x\,\left (4\,x^7-80\,x^6+600\,x^5-2000\,x^4+2520\,x^3-200\,x^2+500\,x\right )+250\,x^2-100\,x^3+635\,x^4-500\,x^5+150\,x^6-20\,x^7+x^8+{\mathrm {e}}^{2\,x}\,\left (6\,x^6-120\,x^5+900\,x^4-3000\,x^3+3760\,x^2-100\,x+250\right )+25} \,d x \]
int(-(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) - exp(x)*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 275*x^2 - 120*x^3 + 638*x^4 - 50 0*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3741*x^2 - 50*x - 2998*x^3 + 90 0*x^4 - 120*x^5 + 6*x^6 + 225) + 20)/(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) - ex p(x)*(500*x - 200*x^2 + 2520*x^3 - 2000*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 250*x^2 - 100*x^3 + 635*x^4 - 500*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)* (3760*x^2 - 100*x - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 250) + 25),x)
int(-(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) - exp(x)*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 275*x^2 - 120*x^3 + 638*x^4 - 50 0*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3741*x^2 - 50*x - 2998*x^3 + 90 0*x^4 - 120*x^5 + 6*x^6 + 225) + 20)/(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) - ex p(x)*(500*x - 200*x^2 + 2520*x^3 - 2000*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 250*x^2 - 100*x^3 + 635*x^4 - 500*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)* (3760*x^2 - 100*x - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 250) + 25), x)