I solved the following non-linear differential equation with mathematica ($x\geq 0$ and $y(0)=p$)
$$\frac{r \left(a-b e^{2 y(x)}+c e^{y(x)}+d e^{\frac{(l+r) y(x)}{l}}\right)}{f-g e^{y(x)}+h e^{\frac{(l+r) y(x)}{l}}}+y'(x)=0$$
the implicit solution for y(x) is
$$\int_1^{y(x)} \frac{g e^{s}-h e^{\frac{(l+r) s}{l}}-f}{-b e^{2 K[1]}+c e^{s}+d e^{\frac{(l+r) s}{l}}+a} \, ds=\int_1^p \frac{g e^{v}-h e^{\frac{(l+r) v}{l}}-f}{-b e^{2 v}+c e^{v}+d e^{\frac{(l+r) v}{l}}+a} \, dv+r x$$
I tried to find some way to obtain an analytical solution to y(x) from this without success. Any clues?