  For fixed $l,m \ge 1$, let
$\mathbf{X}_n^{(0)},\mathbf{X}_n^{(1)},\dots,\mathbf{X}_n^{(l)}$ be independent
random $n \times n$ matrices with independent entries, let $\mathbf{F}_n^{(0)}
:= \mathbf{X}_n^{(0)} (\mathbf{X}_n^{(1)})^{-1} \cdots
(\mathbf{X}_n^{(l)})^{-1}$, and let
$\mathbf{F}_n^{(1)},\dots,\mathbf{F}_n^{(m)}$ be independent random matrices of
the same form as $\mathbf{F}_n^{(0)}$. We investigate the limiting spectral
distributions of the matrices $\mathbf{F}_n^{(0)}$ and $\mathbf{F}_n^{(1)} +
\dots + \mathbf{F}_n^{(m)}$ as $n \to \infty$. Our main result shows that the
sum $\mathbf{F}_n^{(1)} + \dots + \mathbf{F}_n^{(m)}$ has the same limiting
eigenvalue distribution as $\mathbf{F}_n^{(0)}$ after appropriate rescaling.
This extends recent findings by Tikhomirov and Timushev (2014).
  To obtain our results, we apply the general framework recently introduced in
G\"otze, K\"osters and Tikhomirov (2014) to sums of products of independent
random matrices and their inverses. We establish the universality of the
limiting singular value and eigenvalue distributions, and we provide a closer
description of the limiting distributions in terms of free probability theory.
