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, it is also known using Wiles' method that there are no primitive integer solutions, see e.g. the recent results of Siksek-Stoll and Anni-Siksek.

You can find more information in the Wikipedia article on Beal's conjecture and the references therein.

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.