\(\int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} (80 x+80 x^2+600 x^4)+e^x (600 x^2+200 x^3+1000 x^5)}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} (80 x+600 x^4)+e^x (400 x^2+1000 x^5)} \, dx\) [8701]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

x - 100*Defer[Int][(10 + 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3)^(-2), x] - 200*Defer[Int][x/(10 + 4*E^(2*x)*x + 20
*E^x*x^2 + 25*x^3)^2, x] + 200*Defer[Int][(E^x*x^2)/(10 + 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3)^2, x] + 500*Defer
[Int][x^3/(10 + 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3)^2, x] - 200*Defer[Int][(E^x*x^3)/(10 + 4*E^(2*x)*x + 20*E^x
*x^2 + 25*x^3)^2, x] - 500*Defer[Int][x^4/(10 + 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3)^2, x] + 20*Defer[Int][x/(10
+ 4*E^(2*x)*x + 20*E^x*x^2 + 25*x^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (16 e^{4 x} x+160 e^{3 x} x^2+125 x^2 \left (8+5 x^3\right )+200 e^x x \left (3+x+5 x^3\right )+40 e^{2 x} \left (2+2 x+15 x^3\right )\right )}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ & = \int \left (1+\frac {20 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3}-\frac {100 \left (1+2 x-2 e^x x^2-5 x^3+2 e^x x^3+5 x^4\right )}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}\right ) \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \frac {1+2 x-2 e^x x^2-5 x^3+2 e^x x^3+5 x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \left (\frac {1}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {2 x}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}-\frac {2 e^x x^2}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}-\frac {5 x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {2 e^x x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {5 x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}\right ) \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \frac {1}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-200 \int \frac {x}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx+200 \int \frac {e^x x^2}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-200 \int \frac {e^x x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx+500 \int \frac {x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-500 \int \frac {x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 8.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x-\frac {10 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \]

[In]

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*(60
0*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]

[Out]

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

Maple [A] (verified)

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

[In]

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*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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]

[In]

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)+6
25*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+5
00*x^3+100),x, algorithm="fricas")

[Out]

(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 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x - \frac {10 x}{25 x^{3} + 20 x^{2} e^{x} + 4 x e^{2 x} + 10} \]

[In]

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)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]

[In]

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)+6
25*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+5
00*x^3+100),x, algorithm="maxima")

[Out]

(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)

none

Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )} - 10 \, x}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]

[In]

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)+6
25*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+5
00*x^3+100),x, algorithm="giac")

[Out]

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

[In]

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*ex
p(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)

[Out]

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*ex
p(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)