The asymptotically fastest known divide-and-conquer methods for inverting dense structured matrices are essentially variations or extensions of the Morf/Bitmead-Anderson algorithm. Most of them must deal with the growth in length of intermediate generators, and this is done by incorporating various generator compression techniques into the algorithms. One exception is an algorithm by Cardinal, which in the particular case of Cauchy-like matrices avoids such growth by focusing on well-specied, already compressed generators of the inverse. In this paper, we extend Cardinal's method to a broader class of structured matrices including those of Vandermonde, Hankel, and Toeplitz types. Besides, some rst experimental results illustrate the practical interest of the approach.