A search for "partial fractions" in article titles in Mathematical Reviews results in over 100 hits. I have made good use of three articles among them.
From this article I learned about the substitution $x_1^2 = -ax_1 - b$ for handling simple complex roots; see The case of a simple complex root.
Johnston's article, published nine years after Garver's in the same magazine and under the same title, pretty much duplicates Garver's method although it does not refer to it. Actually, Johnston does not claim credit for the technique. He writes:
In manuscript notes left by the late Rear Admiral John P. Merrell, United States Navy, Head of the Department of Mathematics, United States Naval Academy in the 1890's and later (about 1905–1908) President of the Naval War College, the writer discovers the following method of resolving into its partial fractions the proper fraction \[ \frac{f(x)}{(x^2+ax+b)(x^2+cx+d)} \] where the denominator is not separated into linear factors. While it does not appear that the method possesses any advantage on the score of brevity, it is at least somewhat different from the more conventional methods. The method will be illustrated by a particular example rather than proved, though the proof is not difficult.
The idea of using the notation $x$ instead of $x_1$ in The case of a simple complex root is inspired by this article.
From this article I learned the technique of auxiliary fractions for handling repeated complex roots which I have described in The case of repeated complex roots.
The method of auxiliary fractions applies equally well to the case of repeated real roots but the method that I have described in The case of repeated real roots appears to be more efficient for hand calculations.
I have not made use of the methods of this article but I have placed the reference here because of a common idea—what I have called "partial partial fractions expansions", Henrici calls "incomplete decomposition of a rational function into partial fractions".
The focus of Henrici's article is the recurrence relationships among the coefficients of the expansion and their computational complexity. These relationships may be of use as computer algorithms but they are not suited for hand calculations.
Author: Rouben Rostamian