\(\int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} (80 x+80 x^2+600 x^4)+e^x (600 x^2+200 x^3+1000 x^5)}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} (80 x+600 x^4)+e^x (400 x^2+1000 x^5)} \, dx\) [2701]

Optimal result

Integrand size = 136, antiderivative size = 31 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=2 \left (-4+\frac {x}{1+\frac {x+\frac {4}{5 \left (\frac {2 e^x}{5}+x\right )^2}}{x}}\right ) \] Output:

2*x/(1+(4/5/(x+2/5*exp(x))^2+x)/x)-8
 

Mathematica [A] (verified)

Time = 7.90 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x-\frac {10 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \] Input:

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]
 

Output:

x - (10*x)/(10 + 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3)
 

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.

\(\displaystyle \int \frac {625 x^6+160 e^{3 x} x^3+1000 x^3+16 e^{4 x} x^2+e^{2 x} \left (600 x^4+80 x^2+80 x\right )+e^x \left (1000 x^5+200 x^3+600 x^2\right )}{625 x^6+e^{2 x} \left (600 x^4+80 x\right )+160 e^{3 x} x^3+500 x^3+16 e^{4 x} x^2+e^x \left (1000 x^5+400 x^2\right )+100} \, dx\)

\(\Big \downarrow \) 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

\(\displaystyle -100 \int \frac {1}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx-200 \int \frac {x}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+200 \int \frac {e^x x^2}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+500 \int \frac {x^3}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx-200 \int \frac {e^x x^3}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+20 \int \frac {x}{25 x^3+20 e^x x^2+4 e^{2 x} x+10}dx-500 \int \frac {x^4}{\left (25 x^3+20 e^x x^2+4 e^{2 x} x+10\right )^2}dx+x\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.52 (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	\(\frac {100 x^{4}+80 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{2 x} x^{2}}{16 x \,{\mathrm e}^{2 x}+80 \,{\mathrm e}^{x} x^{2}+100 x^{3}+40}\)	\(48\)

Input:

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)
 

Output:

x-10*x/(4*x*exp(x)^2+20*exp(x)*x^2+25*x^3+10)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \] Input:

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="fricas")
 

Output:

(25*x^4 + 20*x^3*e^x + 4*x^2*e^(2*x))/(25*x^3 + 20*x^2*e^x + 4*x*e^(2*x) + 
 10)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x - \frac {10 x}{25 x^{3} + 20 x^{2} e^{x} + 4 x e^{2 x} + 10} \] Input:

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)
 

Output:

x - 10*x/(25*x**3 + 20*x**2*exp(x) + 4*x*exp(2*x) + 10)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \] Input:

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="maxima")
 

Output:

(25*x^4 + 20*x^3*e^x + 4*x^2*e^(2*x))/(25*x^3 + 20*x^2*e^x + 4*x*e^(2*x) + 
 10)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )} - 10 \, x}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \] Input:

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="giac")
 

Output:

(25*x^4 + 20*x^3*e^x + 4*x^2*e^(2*x) - 10*x)/(25*x^3 + 20*x^2*e^x + 4*x*e^ 
(2*x) + 10)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x^2+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+200\,x^3+600\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+1000\,x^3+625\,x^6}{{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+400\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+500\,x^3+625\,x^6+100} \,d x \] Input:

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)
 

Output:

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)
 

Reduce [F]

\[ \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=625 \left (\int \frac {x^{6}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+1000 \left (\int \frac {x^{3}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+16 \left (\int \frac {e^{4 x} x^{2}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+160 \left (\int \frac {e^{3 x} x^{3}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+600 \left (\int \frac {e^{2 x} x^{4}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+80 \left (\int \frac {e^{2 x} x^{2}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+80 \left (\int \frac {e^{2 x} x}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+1000 \left (\int \frac {e^{x} x^{5}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+200 \left (\int \frac {e^{x} x^{3}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right )+600 \left (\int \frac {e^{x} x^{2}}{16 e^{4 x} x^{2}+160 e^{3 x} x^{3}+600 e^{2 x} x^{4}+80 e^{2 x} x +1000 e^{x} x^{5}+400 e^{x} x^{2}+625 x^{6}+500 x^{3}+100}d x \right ) \] Input:

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)
 

Output:

625*int(x**6/(16*e**(4*x)*x**2 + 160*e**(3*x)*x**3 + 600*e**(2*x)*x**4 + 8 
0*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 625*x**6 + 500*x**3 + 100) 
,x) + 1000*int(x**3/(16*e**(4*x)*x**2 + 160*e**(3*x)*x**3 + 600*e**(2*x)*x 
**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 625*x**6 + 500*x**3 
 + 100),x) + 16*int((e**(4*x)*x**2)/(16*e**(4*x)*x**2 + 160*e**(3*x)*x**3 
+ 600*e**(2*x)*x**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 625 
*x**6 + 500*x**3 + 100),x) + 160*int((e**(3*x)*x**3)/(16*e**(4*x)*x**2 + 1 
60*e**(3*x)*x**3 + 600*e**(2*x)*x**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 40 
0*e**x*x**2 + 625*x**6 + 500*x**3 + 100),x) + 600*int((e**(2*x)*x**4)/(16* 
e**(4*x)*x**2 + 160*e**(3*x)*x**3 + 600*e**(2*x)*x**4 + 80*e**(2*x)*x + 10 
00*e**x*x**5 + 400*e**x*x**2 + 625*x**6 + 500*x**3 + 100),x) + 80*int((e** 
(2*x)*x**2)/(16*e**(4*x)*x**2 + 160*e**(3*x)*x**3 + 600*e**(2*x)*x**4 + 80 
*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 625*x**6 + 500*x**3 + 100), 
x) + 80*int((e**(2*x)*x)/(16*e**(4*x)*x**2 + 160*e**(3*x)*x**3 + 600*e**(2 
*x)*x**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 625*x**6 + 500 
*x**3 + 100),x) + 1000*int((e**x*x**5)/(16*e**(4*x)*x**2 + 160*e**(3*x)*x* 
*3 + 600*e**(2*x)*x**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 400*e**x*x**2 + 
625*x**6 + 500*x**3 + 100),x) + 200*int((e**x*x**3)/(16*e**(4*x)*x**2 + 16 
0*e**(3*x)*x**3 + 600*e**(2*x)*x**4 + 80*e**(2*x)*x + 1000*e**x*x**5 + 400 
*e**x*x**2 + 625*x**6 + 500*x**3 + 100),x) + 600*int((e**x*x**2)/(16*e*...