Darmon and Granville proved, using Faltings' Theorem, that your equation has finitely many primitive integer solutions for any fixed exponents which are at least $3$. In fact their result is more general. For several concrete exponents beyond Fermat's Last Theorem, the full set of primitive integer solutions is also known, see e.g. the papers of Siksek-Stoll and Anni-Siksek.
You can find more information in the Wikipedia article on Beal's conjecture and the references therein. See also this survey, especially Section 4.5.