Is there an analytical expression for the real positive solution for $x$ satisfying the equation
$1=\sum_{i=1\ldots N}a_i/({x+b_i})$
assuming $\sum_{i=1\ldots N}a_i/b_i>1$. Being a sum of monotonically decreasing functions, it is easy to see there is exactly one real positive solution for $x$, but is there an expression for it?
No,
because if there were some kind of closed solution, then you could solve any univariate polynomial in this way. For let $p(x)$ be any polynomial of degree $n$, with leading coefficient $1$, and $b_1, ...,b_n$ distinct numbers. Then $$ p(x)=\prod_{k=1}^n(x+b_k)+\sum_{m=1}^n p(-b_m)\prod_{k\ne m}\frac{x+b_k}{b_k-b_m} $$ Any root of $p(x)=0$ would satisfy $$ 1=\sum_{m=1}^n\frac{a_m}{x+b_m} $$ where $$ a_m=-\frac{p(-b_m)}{\prod_{k\ne m}(b_k-b_m)} $$
See Weierstraß' Durand-Kerner method.