Let $f(x)$ be a function:
$f(x) = \sum_{i=1}^{N} \frac{a_i^3}{(1 + xa_i)^3}$
where $a_1,a_2,...,a_n$ are arbitrary constants.
I'd like to find the root of the function $f(x)$. That is, find $x$ where $f(x) = 0$. This is straightforward to solve numerically (I use the Newton method). Does an analytical solution exist for this? I anticipate the answer is "No", but would like a confirmation. Thanks in advance.