So I got something like this for the extra terms:
$$
\sum_{i=1}^{C} \frac1{x-c_i} \left[\prod_{j=1, j\neq i}^{C} \frac1{c_{ij}^2\right]\left[\prod_{j=1}^{B}\frac1{c_i-b_j}\right}\left[-\sum_1^B \frac1{c_i-b_j} + \sum_{1}^C \frac{2(-1)^j}{c_{ij}}\right]
$$