How to simplify $(aX+D)^{-1}$

Question

Given a scale $a$, a full rank, symmetric, off-diagonal matrix $X$ and diagonal matrix $D$

Can the following inverse be simplified so that the scalar $a$ is not included inside the inverse operator?

$(aX+D)^{-1}$

UPDATE:

Actually the inverse of $X$ and $D$ can be obtained, so the question can also be considered as approximating $(aX+D)^{-1}$ by $a$, $X^{-1}$ and $D^{-1}$

Answer 1

There are some results for some special cases. For instance for low-rank updates, see Woodbury expression Question 297799.