Simplification of the ratio between series

Question

I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not pretty to look at, and more importantly, difficult to solve and work with. To cut staight to the point, I am looking either for a way to simplify the following equation: $$y=a\frac{\sum_{i=0}^nx^{n-i}f(i)(-x^2+ibx+c)}{\sum_{i=0}^{n}x^{n-i}f(i)(x^2+dx-c)}$$ Where $$f(i)=f(i-1)\cdot k_i$$ Given a list of $n$ values $k_1,k_2,...k_n\in\mathbb{R}$, and that $f(0)=1$.
I am truly at a loss for how to do this.

(Btw, this formula is "translated" into normal mathematical notation. If anyone is curious about the orginal acid/base-chemistry problem, is it basically the following: Derive a formula that describes a titration curve.)

Litterally any help will be greatly appreciated.