Many pieces of writing are succinct, but they are not rhythmical, giving no pleasure to the ear of the reader. The reason for that is likely this: The writers themselves either are unmindful of rhythm when they write, or possess a poor ability to formulate rhythm. So the question I want to ask is this: How does a writer strengthen their ear for rhythm?
Paying attention to rhythm (i.e., caring about it) is the first step. After that, I recommend reading some instructional material on the subject. A book that has helped me greatly in many areas of writing (not just rhythm) is On Writing Well by William F. Zinsser. It is probably out of print, but if you can get hold of a copy, I highly recommend it.
One more suggestion: read what you have written aloud. Doing so will uncover many problems with rhythm and wording.