Integrand size = 157, antiderivative size = 32 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {\left (2-\frac {-1+x}{-\frac {e^x}{5-4 x}+x}\right )^2}{16 x^8} \] Output:
1/16/x^8*(2-(-1+x)/(x-exp(x)/(-4*x+5)))^2
Time = 8.81 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {\left (5-2 e^x+x-4 x^2\right )^2}{16 x^8 \left (e^x+x (-5+4 x)\right )^2} \] Input:
Integrate[(-16*E^(3*x) + 625*x - 375*x^2 - 1000*x^3 + 640*x^4 + 384*x^5 - 256*x^6 + E^(2*x)*(80 + 124*x - 162*x^2 + 8*x^3) + E^x*(-100 - 560*x + 547 *x^2 + 437*x^3 - 392*x^4 + 16*x^5))/(8*E^(3*x)*x^9 - 1000*x^12 + 2400*x^13 - 1920*x^14 + 512*x^15 + E^(2*x)*(-120*x^10 + 96*x^11) + E^x*(600*x^11 - 960*x^12 + 384*x^13)),x]
Output:
(5 - 2*E^x + x - 4*x^2)^2/(16*x^8*(E^x + x*(-5 + 4*x))^2)
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.
| \(\displaystyle \int \frac {-256 x^6+384 x^5+640 x^4-1000 x^3-375 x^2+e^{2 x} \left (8 x^3-162 x^2+124 x+80\right )+e^x \left (16 x^5-392 x^4+437 x^3+547 x^2-560 x-100\right )+625 x-16 e^{3 x}}{512 x^{15}-1920 x^{14}+2400 x^{13}-1000 x^{12}+8 e^{3 x} x^9+e^{2 x} \left (96 x^{11}-120 x^{10}\right )+e^x \left (384 x^{13}-960 x^{12}+600 x^{11}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-256 x^6+384 x^5+640 x^4-1000 x^3-375 x^2+e^{2 x} \left (8 x^3-162 x^2+124 x+80\right )+e^x \left (16 x^5-392 x^4+437 x^3+547 x^2-560 x-100\right )+625 x-16 e^{3 x}}{8 x^9 \left (4 x^2-5 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int -\frac {256 x^6-384 x^5-640 x^4+1000 x^3+375 x^2-625 x+16 e^{3 x}-2 e^{2 x} \left (4 x^3-81 x^2+62 x+40\right )+e^x \left (-16 x^5+392 x^4-437 x^3-547 x^2+560 x+100\right )}{x^9 \left (4 x^2-5 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{8} \int \frac {256 x^6-384 x^5-640 x^4+1000 x^3+375 x^2-625 x+16 e^{3 x}-2 e^{2 x} \left (4 x^3-81 x^2+62 x+40\right )+e^x \left (-16 x^5+392 x^4-437 x^3-547 x^2+560 x+100\right )}{x^9 \left (4 x^2-5 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{8} \int \left (-\frac {\left (4 x^2-13 x+5\right ) \left (4 x^2-9 x+5\right )^2}{x^8 \left (4 x^2-5 x+e^x\right )^3}-\frac {2 \left (4 x^3+15 x^2-58 x+40\right )}{x^9 \left (4 x^2-5 x+e^x\right )}+\frac {48 x^5-216 x^4+255 x^3+53 x^2-240 x+100}{x^9 \left (4 x^2-5 x+e^x\right )^2}+\frac {16}{x^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (64 \int \frac {1}{x^2 \left (4 x^2-5 x+e^x\right )^3}dx-100 \int \frac {1}{x^9 \left (4 x^2-5 x+e^x\right )^2}dx+80 \int \frac {1}{x^9 \left (4 x^2-5 x+e^x\right )}dx+125 \int \frac {1}{x^8 \left (4 x^2-5 x+e^x\right )^3}dx+240 \int \frac {1}{x^8 \left (4 x^2-5 x+e^x\right )^2}dx-116 \int \frac {1}{x^8 \left (4 x^2-5 x+e^x\right )}dx-775 \int \frac {1}{x^7 \left (4 x^2-5 x+e^x\right )^3}dx-53 \int \frac {1}{x^7 \left (4 x^2-5 x+e^x\right )^2}dx+30 \int \frac {1}{x^7 \left (4 x^2-5 x+e^x\right )}dx+1875 \int \frac {1}{x^6 \left (4 x^2-5 x+e^x\right )^3}dx-255 \int \frac {1}{x^6 \left (4 x^2-5 x+e^x\right )^2}dx+8 \int \frac {1}{x^6 \left (4 x^2-5 x+e^x\right )}dx-2293 \int \frac {1}{x^5 \left (4 x^2-5 x+e^x\right )^3}dx+216 \int \frac {1}{x^5 \left (4 x^2-5 x+e^x\right )^2}dx+1500 \int \frac {1}{x^4 \left (4 x^2-5 x+e^x\right )^3}dx-48 \int \frac {1}{x^4 \left (4 x^2-5 x+e^x\right )^2}dx-496 \int \frac {1}{x^3 \left (4 x^2-5 x+e^x\right )^3}dx+\frac {2}{x^8}\right )\) |
Input:
Int[(-16*E^(3*x) + 625*x - 375*x^2 - 1000*x^3 + 640*x^4 + 384*x^5 - 256*x^ 6 + E^(2*x)*(80 + 124*x - 162*x^2 + 8*x^3) + E^x*(-100 - 560*x + 547*x^2 + 437*x^3 - 392*x^4 + 16*x^5))/(8*E^(3*x)*x^9 - 1000*x^12 + 2400*x^13 - 192 0*x^14 + 512*x^15 + E^(2*x)*(-120*x^10 + 96*x^11) + E^x*(600*x^11 - 960*x^ 12 + 384*x^13)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(29)=58\).
Time = 0.82 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91
| method | result | size |
| risch | \(\frac {16 x^{4}+16 \,{\mathrm e}^{x} x^{2}-8 x^{3}+4 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x -39 x^{2}-20 \,{\mathrm e}^{x}+10 x +25}{16 x^{8} \left (4 x^{2}+{\mathrm e}^{x}-5 x \right )^{2}}\) | \(61\) |
| parallelrisch | \(-\frac {-125-80 x^{4}+40 x^{3}-80 \,{\mathrm e}^{x} x^{2}+195 x^{2}+20 \,{\mathrm e}^{x} x -20 \,{\mathrm e}^{2 x}-50 x +100 \,{\mathrm e}^{x}}{80 x^{8} \left (16 x^{4}-40 x^{3}+8 \,{\mathrm e}^{x} x^{2}+25 x^{2}-10 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}\right )}\) | \(82\) |
Input:
int((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^ 3+547*x^2-560*x-100)*exp(x)-256*x^6+384*x^5+640*x^4-1000*x^3-375*x^2+625*x )/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600*x^11) *exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x,method=_RETURNVERBOSE)
Output:
1/16*(16*x^4+16*exp(x)*x^2-8*x^3+4*exp(2*x)-4*exp(x)*x-39*x^2-20*exp(x)+10 *x+25)/x^8/(4*x^2+exp(x)-5*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.59 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {16 \, x^{4} - 8 \, x^{3} - 39 \, x^{2} + 4 \, {\left (4 \, x^{2} - x - 5\right )} e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 25 \, x^{10} + x^{8} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{10} - 5 \, x^{9}\right )} e^{x}\right )}} \] Input:
integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+ 437*x^3+547*x^2-560*x-100)*exp(x)-256*x^6+384*x^5+640*x^4-1000*x^3-375*x^2 +625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600 *x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="fricas ")
Output:
1/16*(16*x^4 - 8*x^3 - 39*x^2 + 4*(4*x^2 - x - 5)*e^x + 10*x + 4*e^(2*x) + 25)/(16*x^12 - 40*x^11 + 25*x^10 + x^8*e^(2*x) + 2*(4*x^10 - 5*x^9)*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {- 48 x^{4} + 152 x^{3} - 139 x^{2} + 10 x + \left (- 16 x^{2} + 36 x - 20\right ) e^{x} + 25}{256 x^{12} - 640 x^{11} + 400 x^{10} + 16 x^{8} e^{2 x} + \left (128 x^{10} - 160 x^{9}\right ) e^{x}} + \frac {1}{4 x^{8}} \] Input:
integrate((-16*exp(x)**3+(8*x**3-162*x**2+124*x+80)*exp(x)**2+(16*x**5-392 *x**4+437*x**3+547*x**2-560*x-100)*exp(x)-256*x**6+384*x**5+640*x**4-1000* x**3-375*x**2+625*x)/(8*x**9*exp(x)**3+(96*x**11-120*x**10)*exp(x)**2+(384 *x**13-960*x**12+600*x**11)*exp(x)+512*x**15-1920*x**14+2400*x**13-1000*x* *12),x)
Output:
(-48*x**4 + 152*x**3 - 139*x**2 + 10*x + (-16*x**2 + 36*x - 20)*exp(x) + 2 5)/(256*x**12 - 640*x**11 + 400*x**10 + 16*x**8*exp(2*x) + (128*x**10 - 16 0*x**9)*exp(x)) + 1/(4*x**8)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.59 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {16 \, x^{4} - 8 \, x^{3} - 39 \, x^{2} + 4 \, {\left (4 \, x^{2} - x - 5\right )} e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 25 \, x^{10} + x^{8} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{10} - 5 \, x^{9}\right )} e^{x}\right )}} \] Input:
integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+ 437*x^3+547*x^2-560*x-100)*exp(x)-256*x^6+384*x^5+640*x^4-1000*x^3-375*x^2 +625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600 *x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="maxima ")
Output:
1/16*(16*x^4 - 8*x^3 - 39*x^2 + 4*(4*x^2 - x - 5)*e^x + 10*x + 4*e^(2*x) + 25)/(16*x^12 - 40*x^11 + 25*x^10 + x^8*e^(2*x) + 2*(4*x^10 - 5*x^9)*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.62 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {16 \, x^{4} - 8 \, x^{3} + 16 \, x^{2} e^{x} - 39 \, x^{2} - 4 \, x e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} - 20 \, e^{x} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 8 \, x^{10} e^{x} + 25 \, x^{10} - 10 \, x^{9} e^{x} + x^{8} e^{\left (2 \, x\right )}\right )}} \] Input:
integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+ 437*x^3+547*x^2-560*x-100)*exp(x)-256*x^6+384*x^5+640*x^4-1000*x^3-375*x^2 +625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600 *x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="giac")
Output:
1/16*(16*x^4 - 8*x^3 + 16*x^2*e^x - 39*x^2 - 4*x*e^x + 10*x + 4*e^(2*x) - 20*e^x + 25)/(16*x^12 - 40*x^11 + 8*x^10*e^x + 25*x^10 - 10*x^9*e^x + x^8* e^(2*x))
Timed out. \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=-\int \frac {16\,{\mathrm {e}}^{3\,x}-625\,x-{\mathrm {e}}^{2\,x}\,\left (8\,x^3-162\,x^2+124\,x+80\right )+{\mathrm {e}}^x\,\left (-16\,x^5+392\,x^4-437\,x^3-547\,x^2+560\,x+100\right )+375\,x^2+1000\,x^3-640\,x^4-384\,x^5+256\,x^6}{{\mathrm {e}}^x\,\left (384\,x^{13}-960\,x^{12}+600\,x^{11}\right )-{\mathrm {e}}^{2\,x}\,\left (120\,x^{10}-96\,x^{11}\right )+8\,x^9\,{\mathrm {e}}^{3\,x}-1000\,x^{12}+2400\,x^{13}-1920\,x^{14}+512\,x^{15}} \,d x \] Input:
int(-(16*exp(3*x) - 625*x - exp(2*x)*(124*x - 162*x^2 + 8*x^3 + 80) + exp( x)*(560*x - 547*x^2 - 437*x^3 + 392*x^4 - 16*x^5 + 100) + 375*x^2 + 1000*x ^3 - 640*x^4 - 384*x^5 + 256*x^6)/(exp(x)*(600*x^11 - 960*x^12 + 384*x^13) - exp(2*x)*(120*x^10 - 96*x^11) + 8*x^9*exp(3*x) - 1000*x^12 + 2400*x^13 - 1920*x^14 + 512*x^15),x)
Output:
-int((16*exp(3*x) - 625*x - exp(2*x)*(124*x - 162*x^2 + 8*x^3 + 80) + exp( x)*(560*x - 547*x^2 - 437*x^3 + 392*x^4 - 16*x^5 + 100) + 375*x^2 + 1000*x ^3 - 640*x^4 - 384*x^5 + 256*x^6)/(exp(x)*(600*x^11 - 960*x^12 + 384*x^13) - exp(2*x)*(120*x^10 - 96*x^11) + 8*x^9*exp(3*x) - 1000*x^12 + 2400*x^13 - 1920*x^14 + 512*x^15), x)
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.75 \[ \int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx=\frac {4 e^{2 x}+16 e^{x} x^{2}-4 e^{x} x -20 e^{x}+16 x^{4}-8 x^{3}-39 x^{2}+10 x +25}{16 x^{8} \left (e^{2 x}+8 e^{x} x^{2}-10 e^{x} x +16 x^{4}-40 x^{3}+25 x^{2}\right )} \] Input:
int((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^ 3+547*x^2-560*x-100)*exp(x)-256*x^6+384*x^5+640*x^4-1000*x^3-375*x^2+625*x )/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600*x^11) *exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x)
Output:
(4*e**(2*x) + 16*e**x*x**2 - 4*e**x*x - 20*e**x + 16*x**4 - 8*x**3 - 39*x* *2 + 10*x + 25)/(16*x**8*(e**(2*x) + 8*e**x*x**2 - 10*e**x*x + 16*x**4 - 4 0*x**3 + 25*x**2))