Combining algebraic cryptanalysis with other types of cryptanalytic attacks seems to be a worthwhile research topic. Integral cryptanalysis for example can be modeled algebraically, but it remains to be seen whether this approach yields cryptanalytic advances.
In my humble opinion, algebraic cryptanalysis better facilitate the full breach of cryptosystems than other unspecified cryptanalyses. The latter more conveniently chase the revealment of targeted secret keys and credentials.
In principle (I am not a top expert on this aspect) algebraic cryptanalysis expresses the system as an algebraic structure (a polynomial etc.) after some "massage" of the system, and then hopes that under the chosen representation, algebraic operation will simplify the system. Naturally, one thing to consider is to combine it with exhaustive partial search: namely assign some of the key bits (a number of them) and express the rest of the system algebraically. Now, depending on the cost of the algebraic component the size of the partially assigned bits is determined (to balance the search and the algebraic cost). Combinations like this that partially lead part of the system to an algebraic analysis while having some subsystem precomputed (assumed) or leaves some variables after the algebraic search to post analysis (i.e. exhaustive search again) are natural combinations.
This file contains how to do this combination : by exhaustive search and building a table called "S-box" and discuss why the combination is done.
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