I need an algorithm for computing minimal polynomials over the integers of algebraic numbers expressed in radicals that I can implement in a program. The only source I've found so far that gives a concrete algorithm rather than heuristic methods that only sometimes work is this. The method given requires factoring polynomials over fields with finitely-generated algebraic extensions. I think I can see how that problem could be brute-forced in a way that would be guaranteed to terminate, but I did a search to get insight into a more efficient approach. The one Wikipedia gives - Trager's method - seems to require finding exactly the kinds of minimal polynomials I'm trying to find in the first place. Does that mean that users of Trager's method use a different method for finding the minimal polynomials than the one I found? Is there a method that is better than brute force that would be compatible with the method for finding minimal polynomials I found?
As per commenter hardmath's suggestion, an example of the kind of number whose minimal polynomial I'm trying to find: $$\sqrt{2+\sqrt[3]{7}+\sqrt{5}\sqrt[5]{5-\sqrt{3}}}+\sqrt[3]{10}$$ Basically, any expression that consists of a finite, grammatical combination of integers, +, -, *, parentheses, and rational exponentiations.