Integrand size = 275, antiderivative size = 37 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {\frac {\left (5+\frac {2}{-5+\frac {e^x}{3 (-12+x)}}\right )^2}{(-5+x)^2}+x+x^2}{x} \] Output:
(x^2+(5+2/(exp(x)/(3*x-36)-5))^2/(-5+x)^2+x)/x
Time = 6.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.81 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {1}{5-x}+\frac {5}{(-5+x)^2}+\frac {1}{x}+\frac {60 (-12+x)}{\left (180+e^x-15 x\right ) (-5+x)^2 x}+\frac {36 (-12+x)^2}{\left (180+e^x-15 x\right )^2 (-5+x)^2 x}+x \] Input:
Integrate[(617025600 - 524471760*x - 623591460*x^2 + 611580105*x^3 - 21180 3255*x^4 + 37236375*x^5 - 3533625*x^6 + 172125*x^7 - 3375*x^8 + E^(3*x)*(1 25 - 75*x - 125*x^2 + 75*x^3 - 15*x^4 + x^5) + E^(2*x)*(63900 - 47565*x - 63225*x^2 + 46065*x^3 - 11475*x^4 + 1215*x^5 - 45*x^6) + E^x*(10879920 - 8 987112*x - 10753785*x^2 + 9244827*x^3 - 2756547*x^4 + 390825*x^5 - 26325*x ^6 + 675*x^7))/(-729000000*x^2 + 619650000*x^3 - 212017500*x^4 + 37236375* x^5 - 3533625*x^6 + 172125*x^7 - 3375*x^8 + E^(3*x)*(-125*x^2 + 75*x^3 - 1 5*x^4 + x^5) + E^(2*x)*(-67500*x^2 + 46125*x^3 - 11475*x^4 + 1215*x^5 - 45 *x^6) + E^x*(-12150000*x^2 + 9315000*x^3 - 2757375*x^4 + 390825*x^5 - 2632 5*x^6 + 675*x^7)),x]
Output:
(5 - x)^(-1) + 5/(-5 + x)^2 + x^(-1) + (60*(-12 + x))/((180 + E^x - 15*x)* (-5 + x)^2*x) + (36*(-12 + x)^2)/((180 + E^x - 15*x)^2*(-5 + x)^2*x) + 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.
| \(\displaystyle \int \frac {-3375 x^8+172125 x^7-3533625 x^6+37236375 x^5-211803255 x^4+611580105 x^3-623591460 x^2+e^{3 x} \left (x^5-15 x^4+75 x^3-125 x^2-75 x+125\right )+e^{2 x} \left (-45 x^6+1215 x^5-11475 x^4+46065 x^3-63225 x^2-47565 x+63900\right )+e^x \left (675 x^7-26325 x^6+390825 x^5-2756547 x^4+9244827 x^3-10753785 x^2-8987112 x+10879920\right )-524471760 x+617025600}{-3375 x^8+172125 x^7-3533625 x^6+37236375 x^5-212017500 x^4+619650000 x^3-729000000 x^2+e^{3 x} \left (x^5-15 x^4+75 x^3-125 x^2\right )+e^{2 x} \left (-45 x^6+1215 x^5-11475 x^4+46125 x^3-67500 x^2\right )+e^x \left (675 x^7-26325 x^6+390825 x^5-2757375 x^4+9315000 x^3-12150000 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {135 \left (25 x^5-375 x^4+1875 x^3-3125 x^2-1587 x+2645\right ) (x-12)^3-e^{3 x} \left (x^5-15 x^4+75 x^3-125 x^2-75 x+125\right )+15 e^{2 x} \left (3 x^6-81 x^5+765 x^4-3071 x^3+4215 x^2+3171 x-4260\right )-9 e^x \left (75 x^7-2925 x^6+43425 x^5-306283 x^4+1027203 x^3-1194865 x^2-998568 x+1208880\right )}{\left (-15 x+e^x+180\right )^3 (5-x)^3 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {60 \left (x^3-15 x^2+24 x+60\right )}{\left (-15 x+e^x+180\right ) (x-5)^3 x^2}-\frac {36 \left (27 x^4-807 x^3+7510 x^2-20508 x-720\right )}{\left (-15 x+e^x+180\right )^2 (x-5)^3 x^2}+\frac {x^5-15 x^4+75 x^3-125 x^2-75 x+125}{(x-5)^3 x^2}-\frac {1080 (x-13) (x-12)^2}{\left (-15 x+e^x+180\right )^3 (x-5)^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5184}{25} \int \frac {1}{\left (-15 x+e^x+180\right )^2 x^2}dx+\frac {144}{5} \int \frac {1}{\left (-15 x+e^x+180\right ) x^2}dx-1080 \int \frac {1}{\left (-15 x+e^x+180\right )^3}dx-\frac {3528}{5} \int \frac {1}{\left (-15 x+e^x+180\right )^2 (x-5)^3}dx+168 \int \frac {1}{\left (-15 x+e^x+180\right ) (x-5)^3}dx+84672 \int \frac {1}{\left (-15 x+e^x+180\right )^3 (x-5)^2}dx-\frac {265356}{25} \int \frac {1}{\left (-15 x+e^x+180\right )^2 (x-5)^2}dx+\frac {276}{5} \int \frac {1}{\left (-15 x+e^x+180\right ) (x-5)^2}dx-\frac {258552}{5} \int \frac {1}{\left (-15 x+e^x+180\right )^3 (x-5)}dx+\frac {126468}{25} \int \frac {1}{\left (-15 x+e^x+180\right )^2 (x-5)}dx-\frac {144}{5} \int \frac {1}{\left (-15 x+e^x+180\right ) (x-5)}dx+\frac {404352}{5} \int \frac {1}{\left (-15 x+e^x+180\right )^3 x}dx-\frac {150768}{25} \int \frac {1}{\left (-15 x+e^x+180\right )^2 x}dx+\frac {144}{5} \int \frac {1}{\left (-15 x+e^x+180\right ) x}dx+x+\frac {1}{5-x}+\frac {5}{(5-x)^2}+\frac {1}{x}\) |
Input:
Int[(617025600 - 524471760*x - 623591460*x^2 + 611580105*x^3 - 211803255*x ^4 + 37236375*x^5 - 3533625*x^6 + 172125*x^7 - 3375*x^8 + E^(3*x)*(125 - 7 5*x - 125*x^2 + 75*x^3 - 15*x^4 + x^5) + E^(2*x)*(63900 - 47565*x - 63225* x^2 + 46065*x^3 - 11475*x^4 + 1215*x^5 - 45*x^6) + E^x*(10879920 - 8987112 *x - 10753785*x^2 + 9244827*x^3 - 2756547*x^4 + 390825*x^5 - 26325*x^6 + 6 75*x^7))/(-729000000*x^2 + 619650000*x^3 - 212017500*x^4 + 37236375*x^5 - 3533625*x^6 + 172125*x^7 - 3375*x^8 + E^(3*x)*(-125*x^2 + 75*x^3 - 15*x^4 + x^5) + E^(2*x)*(-67500*x^2 + 46125*x^3 - 11475*x^4 + 1215*x^5 - 45*x^6) + E^x*(-12150000*x^2 + 9315000*x^3 - 2757375*x^4 + 390825*x^5 - 26325*x^6 + 675*x^7)),x]
Output:
$Aborted
Time = 2.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(x +\frac {25}{x \left (x^{2}-10 x +25\right )}-\frac {12 \left (72 x^{2}-5 \,{\mathrm e}^{x} x -1728 x +60 \,{\mathrm e}^{x}+10368\right )}{x \left (x^{2}-10 x +25\right ) \left (15 x -{\mathrm e}^{x}-180\right )^{2}}\) | \(63\) |
| parallelrisch | \(\frac {685584+7985736 x +360 \,{\mathrm e}^{x} x^{4}-34500 \,{\mathrm e}^{x} x^{2}+89310 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x} x^{4}-75 \,{\mathrm e}^{2 x} x^{2}+2250 \,{\mathrm e}^{x} x^{3}-30 x^{5} {\mathrm e}^{x}+250 x \,{\mathrm e}^{2 x}+225 x^{6}-5400 x^{5}-3775239 x^{2}+8280 \,{\mathrm e}^{x}+461250 x^{3}+15525 x^{4}+25 \,{\mathrm e}^{2 x}}{x \left (225 x^{4}-30 \,{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}-7650 x^{3}+660 \,{\mathrm e}^{x} x^{2}-10 x \,{\mathrm e}^{2 x}+92025 x^{2}-4350 \,{\mathrm e}^{x} x +25 \,{\mathrm e}^{2 x}-459000 x +9000 \,{\mathrm e}^{x}+810000\right )}\) | \(168\) |
Input:
int(((x^5-15*x^4+75*x^3-125*x^2-75*x+125)*exp(x)^3+(-45*x^6+1215*x^5-11475 *x^4+46065*x^3-63225*x^2-47565*x+63900)*exp(x)^2+(675*x^7-26325*x^6+390825 *x^5-2756547*x^4+9244827*x^3-10753785*x^2-8987112*x+10879920)*exp(x)-3375* x^8+172125*x^7-3533625*x^6+37236375*x^5-211803255*x^4+611580105*x^3-623591 460*x^2-524471760*x+617025600)/((x^5-15*x^4+75*x^3-125*x^2)*exp(x)^3+(-45* x^6+1215*x^5-11475*x^4+46125*x^3-67500*x^2)*exp(x)^2+(675*x^7-26325*x^6+39 0825*x^5-2757375*x^4+9315000*x^3-12150000*x^2)*exp(x)-3375*x^8+172125*x^7- 3533625*x^6+37236375*x^5-212017500*x^4+619650000*x^3-729000000*x^2),x,meth od=_RETURNVERBOSE)
Output:
x+25/x/(x^2-10*x+25)-12/x*(72*x^2-5*exp(x)*x-1728*x+60*exp(x)+10368)/(x^2- 10*x+25)/(15*x-exp(x)-180)^2
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (33) = 66\).
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.84 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {225 \, x^{6} - 7650 \, x^{5} + 92025 \, x^{4} - 459000 \, x^{3} + 814761 \, x^{2} + {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 30 \, {\left (x^{5} - 22 \, x^{4} + 145 \, x^{3} - 300 \, x^{2} + 23 \, x - 276\right )} e^{x} - 114264 \, x + 685584}{225 \, x^{5} - 7650 \, x^{4} + 92025 \, x^{3} - 459000 \, x^{2} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 30 \, {\left (x^{4} - 22 \, x^{3} + 145 \, x^{2} - 300 \, x\right )} e^{x} + 810000 \, x} \] Input:
integrate(((x^5-15*x^4+75*x^3-125*x^2-75*x+125)*exp(x)^3+(-45*x^6+1215*x^5 -11475*x^4+46065*x^3-63225*x^2-47565*x+63900)*exp(x)^2+(675*x^7-26325*x^6+ 390825*x^5-2756547*x^4+9244827*x^3-10753785*x^2-8987112*x+10879920)*exp(x) -3375*x^8+172125*x^7-3533625*x^6+37236375*x^5-211803255*x^4+611580105*x^3- 623591460*x^2-524471760*x+617025600)/((x^5-15*x^4+75*x^3-125*x^2)*exp(x)^3 +(-45*x^6+1215*x^5-11475*x^4+46125*x^3-67500*x^2)*exp(x)^2+(675*x^7-26325* x^6+390825*x^5-2757375*x^4+9315000*x^3-12150000*x^2)*exp(x)-3375*x^8+17212 5*x^7-3533625*x^6+37236375*x^5-212017500*x^4+619650000*x^3-729000000*x^2), x, algorithm="fricas")
Output:
(225*x^6 - 7650*x^5 + 92025*x^4 - 459000*x^3 + 814761*x^2 + (x^4 - 10*x^3 + 25*x^2 + 25)*e^(2*x) - 30*(x^5 - 22*x^4 + 145*x^3 - 300*x^2 + 23*x - 276 )*e^x - 114264*x + 685584)/(225*x^5 - 7650*x^4 + 92025*x^3 - 459000*x^2 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 30*(x^4 - 22*x^3 + 145*x^2 - 300*x)*e^x + 810000*x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.22 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=x + \frac {- \frac {2 x^{2}}{3} + 16 x + \left (\frac {5 x}{108} - \frac {5}{9}\right ) e^{x} - 96}{\frac {25 x^{5}}{144} - \frac {425 x^{4}}{72} + \frac {10225 x^{3}}{144} - \frac {2125 x^{2}}{6} + 625 x + \left (\frac {x^{3}}{1296} - \frac {5 x^{2}}{648} + \frac {25 x}{1296}\right ) e^{2 x} + \left (- \frac {5 x^{4}}{216} + \frac {55 x^{3}}{108} - \frac {725 x^{2}}{216} + \frac {125 x}{18}\right ) e^{x}} + \frac {25}{x^{3} - 10 x^{2} + 25 x} \] Input:
integrate(((x**5-15*x**4+75*x**3-125*x**2-75*x+125)*exp(x)**3+(-45*x**6+12 15*x**5-11475*x**4+46065*x**3-63225*x**2-47565*x+63900)*exp(x)**2+(675*x** 7-26325*x**6+390825*x**5-2756547*x**4+9244827*x**3-10753785*x**2-8987112*x +10879920)*exp(x)-3375*x**8+172125*x**7-3533625*x**6+37236375*x**5-2118032 55*x**4+611580105*x**3-623591460*x**2-524471760*x+617025600)/((x**5-15*x** 4+75*x**3-125*x**2)*exp(x)**3+(-45*x**6+1215*x**5-11475*x**4+46125*x**3-67 500*x**2)*exp(x)**2+(675*x**7-26325*x**6+390825*x**5-2757375*x**4+9315000* x**3-12150000*x**2)*exp(x)-3375*x**8+172125*x**7-3533625*x**6+37236375*x** 5-212017500*x**4+619650000*x**3-729000000*x**2),x)
Output:
x + (-2*x**2/3 + 16*x + (5*x/108 - 5/9)*exp(x) - 96)/(25*x**5/144 - 425*x* *4/72 + 10225*x**3/144 - 2125*x**2/6 + 625*x + (x**3/1296 - 5*x**2/648 + 2 5*x/1296)*exp(2*x) + (-5*x**4/216 + 55*x**3/108 - 725*x**2/216 + 125*x/18) *exp(x)) + 25/(x**3 - 10*x**2 + 25*x)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (33) = 66\).
Time = 0.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.84 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {225 \, x^{6} - 7650 \, x^{5} + 92025 \, x^{4} - 459000 \, x^{3} + 814761 \, x^{2} + {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 30 \, {\left (x^{5} - 22 \, x^{4} + 145 \, x^{3} - 300 \, x^{2} + 23 \, x - 276\right )} e^{x} - 114264 \, x + 685584}{225 \, x^{5} - 7650 \, x^{4} + 92025 \, x^{3} - 459000 \, x^{2} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 30 \, {\left (x^{4} - 22 \, x^{3} + 145 \, x^{2} - 300 \, x\right )} e^{x} + 810000 \, x} \] Input:
integrate(((x^5-15*x^4+75*x^3-125*x^2-75*x+125)*exp(x)^3+(-45*x^6+1215*x^5 -11475*x^4+46065*x^3-63225*x^2-47565*x+63900)*exp(x)^2+(675*x^7-26325*x^6+ 390825*x^5-2756547*x^4+9244827*x^3-10753785*x^2-8987112*x+10879920)*exp(x) -3375*x^8+172125*x^7-3533625*x^6+37236375*x^5-211803255*x^4+611580105*x^3- 623591460*x^2-524471760*x+617025600)/((x^5-15*x^4+75*x^3-125*x^2)*exp(x)^3 +(-45*x^6+1215*x^5-11475*x^4+46125*x^3-67500*x^2)*exp(x)^2+(675*x^7-26325* x^6+390825*x^5-2757375*x^4+9315000*x^3-12150000*x^2)*exp(x)-3375*x^8+17212 5*x^7-3533625*x^6+37236375*x^5-212017500*x^4+619650000*x^3-729000000*x^2), x, algorithm="maxima")
Output:
(225*x^6 - 7650*x^5 + 92025*x^4 - 459000*x^3 + 814761*x^2 + (x^4 - 10*x^3 + 25*x^2 + 25)*e^(2*x) - 30*(x^5 - 22*x^4 + 145*x^3 - 300*x^2 + 23*x - 276 )*e^x - 114264*x + 685584)/(225*x^5 - 7650*x^4 + 92025*x^3 - 459000*x^2 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 30*(x^4 - 22*x^3 + 145*x^2 - 300*x)*e^x + 810000*x)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (33) = 66\).
Time = 0.17 (sec) , antiderivative size = 176, normalized size of antiderivative = 4.76 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {225 \, x^{6} - 30 \, x^{5} e^{x} - 7650 \, x^{5} + x^{4} e^{\left (2 \, x\right )} + 660 \, x^{4} e^{x} + 92025 \, x^{4} - 10 \, x^{3} e^{\left (2 \, x\right )} - 4350 \, x^{3} e^{x} - 459000 \, x^{3} + 25 \, x^{2} e^{\left (2 \, x\right )} + 9000 \, x^{2} e^{x} + 814761 \, x^{2} - 690 \, x e^{x} - 114264 \, x + 25 \, e^{\left (2 \, x\right )} + 8280 \, e^{x} + 685584}{225 \, x^{5} - 30 \, x^{4} e^{x} - 7650 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 660 \, x^{3} e^{x} + 92025 \, x^{3} - 10 \, x^{2} e^{\left (2 \, x\right )} - 4350 \, x^{2} e^{x} - 459000 \, x^{2} + 25 \, x e^{\left (2 \, x\right )} + 9000 \, x e^{x} + 810000 \, x} \] Input:
integrate(((x^5-15*x^4+75*x^3-125*x^2-75*x+125)*exp(x)^3+(-45*x^6+1215*x^5 -11475*x^4+46065*x^3-63225*x^2-47565*x+63900)*exp(x)^2+(675*x^7-26325*x^6+ 390825*x^5-2756547*x^4+9244827*x^3-10753785*x^2-8987112*x+10879920)*exp(x) -3375*x^8+172125*x^7-3533625*x^6+37236375*x^5-211803255*x^4+611580105*x^3- 623591460*x^2-524471760*x+617025600)/((x^5-15*x^4+75*x^3-125*x^2)*exp(x)^3 +(-45*x^6+1215*x^5-11475*x^4+46125*x^3-67500*x^2)*exp(x)^2+(675*x^7-26325* x^6+390825*x^5-2757375*x^4+9315000*x^3-12150000*x^2)*exp(x)-3375*x^8+17212 5*x^7-3533625*x^6+37236375*x^5-212017500*x^4+619650000*x^3-729000000*x^2), x, algorithm="giac")
Output:
(225*x^6 - 30*x^5*e^x - 7650*x^5 + x^4*e^(2*x) + 660*x^4*e^x + 92025*x^4 - 10*x^3*e^(2*x) - 4350*x^3*e^x - 459000*x^3 + 25*x^2*e^(2*x) + 9000*x^2*e^ x + 814761*x^2 - 690*x*e^x - 114264*x + 25*e^(2*x) + 8280*e^x + 685584)/(2 25*x^5 - 30*x^4*e^x - 7650*x^4 + x^3*e^(2*x) + 660*x^3*e^x + 92025*x^3 - 1 0*x^2*e^(2*x) - 4350*x^2*e^x - 459000*x^2 + 25*x*e^(2*x) + 9000*x*e^x + 81 0000*x)
Timed out. \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\int \frac {524471760\,x+{\mathrm {e}}^{3\,x}\,\left (-x^5+15\,x^4-75\,x^3+125\,x^2+75\,x-125\right )+{\mathrm {e}}^x\,\left (-675\,x^7+26325\,x^6-390825\,x^5+2756547\,x^4-9244827\,x^3+10753785\,x^2+8987112\,x-10879920\right )+623591460\,x^2-611580105\,x^3+211803255\,x^4-37236375\,x^5+3533625\,x^6-172125\,x^7+3375\,x^8+{\mathrm {e}}^{2\,x}\,\left (45\,x^6-1215\,x^5+11475\,x^4-46065\,x^3+63225\,x^2+47565\,x-63900\right )-617025600}{{\mathrm {e}}^{2\,x}\,\left (45\,x^6-1215\,x^5+11475\,x^4-46125\,x^3+67500\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (-x^5+15\,x^4-75\,x^3+125\,x^2\right )+729000000\,x^2-619650000\,x^3+212017500\,x^4-37236375\,x^5+3533625\,x^6-172125\,x^7+3375\,x^8+{\mathrm {e}}^x\,\left (-675\,x^7+26325\,x^6-390825\,x^5+2757375\,x^4-9315000\,x^3+12150000\,x^2\right )} \,d x \] Input:
int((524471760*x + exp(3*x)*(75*x + 125*x^2 - 75*x^3 + 15*x^4 - x^5 - 125) + exp(x)*(8987112*x + 10753785*x^2 - 9244827*x^3 + 2756547*x^4 - 390825*x ^5 + 26325*x^6 - 675*x^7 - 10879920) + 623591460*x^2 - 611580105*x^3 + 211 803255*x^4 - 37236375*x^5 + 3533625*x^6 - 172125*x^7 + 3375*x^8 + exp(2*x) *(47565*x + 63225*x^2 - 46065*x^3 + 11475*x^4 - 1215*x^5 + 45*x^6 - 63900) - 617025600)/(exp(2*x)*(67500*x^2 - 46125*x^3 + 11475*x^4 - 1215*x^5 + 45 *x^6) + exp(3*x)*(125*x^2 - 75*x^3 + 15*x^4 - x^5) + 729000000*x^2 - 61965 0000*x^3 + 212017500*x^4 - 37236375*x^5 + 3533625*x^6 - 172125*x^7 + 3375* x^8 + exp(x)*(12150000*x^2 - 9315000*x^3 + 2757375*x^4 - 390825*x^5 + 2632 5*x^6 - 675*x^7)),x)
Output:
int((524471760*x + exp(3*x)*(75*x + 125*x^2 - 75*x^3 + 15*x^4 - x^5 - 125) + exp(x)*(8987112*x + 10753785*x^2 - 9244827*x^3 + 2756547*x^4 - 390825*x ^5 + 26325*x^6 - 675*x^7 - 10879920) + 623591460*x^2 - 611580105*x^3 + 211 803255*x^4 - 37236375*x^5 + 3533625*x^6 - 172125*x^7 + 3375*x^8 + exp(2*x) *(47565*x + 63225*x^2 - 46065*x^3 + 11475*x^4 - 1215*x^5 + 45*x^6 - 63900) - 617025600)/(exp(2*x)*(67500*x^2 - 46125*x^3 + 11475*x^4 - 1215*x^5 + 45 *x^6) + exp(3*x)*(125*x^2 - 75*x^3 + 15*x^4 - x^5) + 729000000*x^2 - 61965 0000*x^3 + 212017500*x^4 - 37236375*x^5 + 3533625*x^6 - 172125*x^7 + 3375* x^8 + exp(x)*(12150000*x^2 - 9315000*x^3 + 2757375*x^4 - 390825*x^5 + 2632 5*x^6 - 675*x^7)), x)
Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.03 \[ \int \frac {617025600-524471760 x-623591460 x^2+611580105 x^3-211803255 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (125-75 x-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (63900-47565 x-63225 x^2+46065 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (10879920-8987112 x-10753785 x^2+9244827 x^3-2756547 x^4+390825 x^5-26325 x^6+675 x^7\right )}{-729000000 x^2+619650000 x^3-212017500 x^4+37236375 x^5-3533625 x^6+172125 x^7-3375 x^8+e^{3 x} \left (-125 x^2+75 x^3-15 x^4+x^5\right )+e^{2 x} \left (-67500 x^2+46125 x^3-11475 x^4+1215 x^5-45 x^6\right )+e^x \left (-12150000 x^2+9315000 x^3-2757375 x^4+390825 x^5-26325 x^6+675 x^7\right )} \, dx=\frac {2 e^{2 x} x^{4}-15 e^{2 x} x^{3}+125 e^{2 x} x +50 e^{2 x}-60 e^{x} x^{5}+1170 e^{x} x^{4}-5400 e^{x} x^{3}-3750 e^{x} x^{2}+43620 e^{x} x +16560 e^{x}+450 x^{6}-14175 x^{5}+145800 x^{4}-457875 x^{3}-665478 x^{2}+3821472 x +1371168}{2 x \left (e^{2 x} x^{2}-10 e^{2 x} x +25 e^{2 x}-30 e^{x} x^{3}+660 e^{x} x^{2}-4350 e^{x} x +9000 e^{x}+225 x^{4}-7650 x^{3}+92025 x^{2}-459000 x +810000\right )} \] Input:
int(((x^5-15*x^4+75*x^3-125*x^2-75*x+125)*exp(x)^3+(-45*x^6+1215*x^5-11475 *x^4+46065*x^3-63225*x^2-47565*x+63900)*exp(x)^2+(675*x^7-26325*x^6+390825 *x^5-2756547*x^4+9244827*x^3-10753785*x^2-8987112*x+10879920)*exp(x)-3375* x^8+172125*x^7-3533625*x^6+37236375*x^5-211803255*x^4+611580105*x^3-623591 460*x^2-524471760*x+617025600)/((x^5-15*x^4+75*x^3-125*x^2)*exp(x)^3+(-45* x^6+1215*x^5-11475*x^4+46125*x^3-67500*x^2)*exp(x)^2+(675*x^7-26325*x^6+39 0825*x^5-2757375*x^4+9315000*x^3-12150000*x^2)*exp(x)-3375*x^8+172125*x^7- 3533625*x^6+37236375*x^5-212017500*x^4+619650000*x^3-729000000*x^2),x)
Output:
(2*e**(2*x)*x**4 - 15*e**(2*x)*x**3 + 125*e**(2*x)*x + 50*e**(2*x) - 60*e* *x*x**5 + 1170*e**x*x**4 - 5400*e**x*x**3 - 3750*e**x*x**2 + 43620*e**x*x + 16560*e**x + 450*x**6 - 14175*x**5 + 145800*x**4 - 457875*x**3 - 665478* x**2 + 3821472*x + 1371168)/(2*x*(e**(2*x)*x**2 - 10*e**(2*x)*x + 25*e**(2 *x) - 30*e**x*x**3 + 660*e**x*x**2 - 4350*e**x*x + 9000*e**x + 225*x**4 - 7650*x**3 + 92025*x**2 - 459000*x + 810000))