Integrand size = 136, antiderivative size = 31 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=2 \left (-4+\frac {x}{1+\frac {x+\frac {4}{5 \left (\frac {2 e^x}{5}+x\right )^2}}{x}}\right ) \]
Time = 8.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x-\frac {10 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \]
Integrate[(16*E^(4*x)*x^2 + 1000*x^3 + 160*E^(3*x)*x^3 + 625*x^6 + E^(2*x) *(80*x + 80*x^2 + 600*x^4) + E^x*(600*x^2 + 200*x^3 + 1000*x^5))/(100 + 16 *E^(4*x)*x^2 + 500*x^3 + 160*E^(3*x)*x^3 + 625*x^6 + E^(2*x)*(80*x + 600*x ^4) + E^x*(400*x^2 + 1000*x^5)),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.
\(\displaystyle \int \frac {625 x^6+160 e^{3 x} x^3+1000 x^3+16 e^{4 x} x^2+e^{2 x} \left (600 x^4+80 x^2+80 x\right )+e^x \left (1000 x^5+200 x^3+600 x^2\right )}{625 x^6+e^{2 x} \left (600 x^4+80 x\right )+160 e^{3 x} x^3+500 x^3+16 e^{4 x} x^2+e^x \left (1000 x^5+400 x^2\right )+100} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {625 x^6+160 e^{3 x} x^3+1000 x^3+16 e^{4 x} x^2+e^{2 x} \left (600 x^4+80 x^2+80 x\right )+e^x \left (1000 x^5+200 x^3+600 x^2\right )}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {20 x}{25 x^3+20 e^x x^2+4 e^{2 x} x+10}-\frac {100 \left (5 x^4+2 e^x x^3-5 x^3-2 e^x x^2+2 x+1\right )}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -100 \int \frac {1}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx-200 \int \frac {x}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+200 \int \frac {e^x x^2}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+500 \int \frac {x^3}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx-200 \int \frac {e^x x^3}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+20 \int \frac {x}{25 x^3+20 e^x x^2+4 e^{2 x} x+10}dx-500 \int \frac {x^4}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+x\) |
Int[(16*E^(4*x)*x^2 + 1000*x^3 + 160*E^(3*x)*x^3 + 625*x^6 + E^(2*x)*(80*x + 80*x^2 + 600*x^4) + E^x*(600*x^2 + 200*x^3 + 1000*x^5))/(100 + 16*E^(4* x)*x^2 + 500*x^3 + 160*E^(3*x)*x^3 + 625*x^6 + E^(2*x)*(80*x + 600*x^4) + E^x*(400*x^2 + 1000*x^5)),x]
3.28.1.3.1 Defintions of rubi rules used
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
risch | \(x -\frac {10 x}{4 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}+25 x^{3}+10}\) | \(29\) |
parallelrisch | \(\frac {100 x^{4}+80 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{2 x} x^{2}}{16 x \,{\mathrm e}^{2 x}+80 \,{\mathrm e}^{x} x^{2}+100 x^{3}+40}\) | \(48\) |
int((16*x^2*exp(x)^4+160*x^3*exp(x)^3+(600*x^4+80*x^2+80*x)*exp(x)^2+(1000 *x^5+200*x^3+600*x^2)*exp(x)+625*x^6+1000*x^3)/(16*x^2*exp(x)^4+160*x^3*ex p(x)^3+(600*x^4+80*x)*exp(x)^2+(1000*x^5+400*x^2)*exp(x)+625*x^6+500*x^3+1 00),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
integrate((16*x^2*exp(x)^4+160*x^3*exp(x)^3+(600*x^4+80*x^2+80*x)*exp(x)^2 +(1000*x^5+200*x^3+600*x^2)*exp(x)+625*x^6+1000*x^3)/(16*x^2*exp(x)^4+160* x^3*exp(x)^3+(600*x^4+80*x)*exp(x)^2+(1000*x^5+400*x^2)*exp(x)+625*x^6+500 *x^3+100),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x - \frac {10 x}{25 x^{3} + 20 x^{2} e^{x} + 4 x e^{2 x} + 10} \]
integrate((16*x**2*exp(x)**4+160*x**3*exp(x)**3+(600*x**4+80*x**2+80*x)*ex p(x)**2+(1000*x**5+200*x**3+600*x**2)*exp(x)+625*x**6+1000*x**3)/(16*x**2* exp(x)**4+160*x**3*exp(x)**3+(600*x**4+80*x)*exp(x)**2+(1000*x**5+400*x**2 )*exp(x)+625*x**6+500*x**3+100),x)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
integrate((16*x^2*exp(x)^4+160*x^3*exp(x)^3+(600*x^4+80*x^2+80*x)*exp(x)^2 +(1000*x^5+200*x^3+600*x^2)*exp(x)+625*x^6+1000*x^3)/(16*x^2*exp(x)^4+160* x^3*exp(x)^3+(600*x^4+80*x)*exp(x)^2+(1000*x^5+400*x^2)*exp(x)+625*x^6+500 *x^3+100),x, algorithm=\
Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )} - 10 \, x}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
integrate((16*x^2*exp(x)^4+160*x^3*exp(x)^3+(600*x^4+80*x^2+80*x)*exp(x)^2 +(1000*x^5+200*x^3+600*x^2)*exp(x)+625*x^6+1000*x^3)/(16*x^2*exp(x)^4+160* x^3*exp(x)^3+(600*x^4+80*x)*exp(x)^2+(1000*x^5+400*x^2)*exp(x)+625*x^6+500 *x^3+100),x, algorithm=\
Timed out. \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x^2+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+200\,x^3+600\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+1000\,x^3+625\,x^6}{{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+400\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+500\,x^3+625\,x^6+100} \,d x \]
int((exp(2*x)*(80*x + 80*x^2 + 600*x^4) + exp(x)*(600*x^2 + 200*x^3 + 1000 *x^5) + 16*x^2*exp(4*x) + 160*x^3*exp(3*x) + 1000*x^3 + 625*x^6)/(exp(2*x) *(80*x + 600*x^4) + exp(x)*(400*x^2 + 1000*x^5) + 16*x^2*exp(4*x) + 160*x^ 3*exp(3*x) + 500*x^3 + 625*x^6 + 100),x)