[Darmon and Granville][1] proved, using [Faltings' Theorem][2], 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][3] and [Anni-Siksek][4].

You can find more information in the [Wikipedia article on Beal's conjecture][5] and the references therein. See also [this survey][6], especially Section 4.5.


  [1]: https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms/27.6.513
  [2]: https://en.wikipedia.org/wiki/Faltings%27s_theorem
  [3]: https://arxiv.org/abs/1309.4421
  [4]: https://arxiv.org/abs/1506.02860
  [5]: https://en.wikipedia.org/wiki/Beal_conjecture
  [6]: https://homepages.warwick.ac.uk/~maseap/papers/bealconj.pdf