Is there a unified formula for summation of fractions of polynomials with only even powers in the denominator?

Question

I know that using the residue theorem we can write: $\frac{1}{\sum_{i=1}^{L}\frac{a_i}{b_i+c_ix^2}}=(\sum_{i=1}^{L-1}\frac{d_i}{x^2+e_i})+K(x)$ where $a_i>=0, b_i>0, c_i>0, d_i<=0, e_i>0$, $K(x)=x^2+s$, and $d_i, e_i, s$ are all functions of $a_i, b_i, c_i, L, i=1,...,L$. I need to know if there is a unified formula for $d_i, e_i, s$ in terms of $a_i, b_i, c_i, L, i=1,...,L$. Thank you in advance.