Using matrix inversion and summation together. How to compute $(A+B^{-1})^{-1}|xrangle$

I need to compute $(A+B^{-1})^{-1}|xrangle$, where A and B is hermitian for simplicity.

I read out about matrix inversion algorithms (HHL, QSVE,…), and algorithms of summing up of matrices(LCU, Lie–Trotter product formula,…) but I don’t understand how to glue this two parts together.