I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: $$x+r\frac{e}{e+t+x^2u}+\frac{xr^2\frac{e}{e+t+x^2u}}{1+r\frac{e}{e+t+x^2u}}=e,$$ which after rearranging yields $$x\left(1+r\frac{e}{e+t+x^2u}\right)+r\frac{e}{e+t+x^2u}\left(1+r\frac{e}{e+t+x^2u}\right)+xr^2\frac{e}{e+t+x^2u}=e\left(1+r\frac{e}{e+t+x^2u}\right).$$ Multiplying by the denominators (i.e., multiplying by $(e+t+x^2u)^2)$, this yields a 5th degree polynomial in $x$. The good news is that a 5th degree polynomial always has at least one solution in the real numbers. The bad news is that there may not be an explicit expression for the solution(s), and even if there were, one could not really work with them (too long/complicated).
Given this problem:
Is there a simpler/better way to find solutions? Some trick you can think of? Since many terms appear repeatedly, maybe there is some potential for simplification.
If not, is there a nice way to approximate the condition, say by a 2nd degree Taylor approximation, and then solve it? My problem with the Taylor approximation is that the term has to be approximated at some point $x^*$, which ideally is the solution, but since I want to make the approximation to find a solution in the first place, this seems difficult. So at which point would I approximate?
All I am able to do right now is find solutions for special cases. For example, $e\to 0$ leads to $x=0$. But these are really just special cases.
Ideally, I would like to have a solution (or solutions) that can be expressed as some explicit function of the parameters $(e,u,r,t)$, i.e., some $x^*=f(e,u,r,t)$. The $f$ should be simple enough so that one can see how the parameters $(e,u,r,t)$ change the solution (e.g., if $e$ increases, then the solution $x^*$ decreases). To achieve this, an approximation is fine.
Any help is very much appreciated! Thanks!
For example, if $r, t, u$ are small, so that we write $r = R\epsilon$, $t = T\epsilon$, $u =U\epsilon$, there is a series solution that starts $$ x = e-R\epsilon-{\frac {R \left( {e}^{2}R-U{e}^{2}-T \right) }{e}}{ \epsilon}^{2}++{\frac {R \left( RU{e}^{4}-{U}^{2}{e}^{4}+{R}^{2}{e}^{3} +{R}^{2}{e}^{2}+{e}^{2}RT-2\,RU{e}^{2}-2\,TU{e}^{2}-{T}^{2} \right) }{ {e}^{2}}}{\epsilon}^{3} +\ldots $$