During my research I came across the following problem:

I need to find a root of the following function: $$\Gamma_{N}(x) = \sum\limits_{i=0}^{M}\left(\frac{\sum\limits_{n=0}^{n_F}n\ \alpha_{i,n} x^n}{\sum\limits_{n=0}^{n_F}\alpha_{i,n} x^n}\right)- N,$$ where $\alpha_{i,n}$ are positive coefficients such that in general, $\alpha_{i,n} \in [0,1]$. Parameter $M$ and $n_F$ are integer numbers and $N$ is real. I want to consider only positive $x$, so I am searching for root in the domain $x \in [0, +\infty]$. I will do all calculations numerically, but I was wondering if this function possesses some properties?

Question: Is this function monotonic, such that there exists only a single root?