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

You can find more information in the [Wikipedia article on Beal's conjecture][5] and the references therein. 


  [1]: https://en.wikipedia.org/wiki/Faltings%27s_theorem
  [2]: https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms/27.6.513
  [3]: https://arxiv.org/abs/1309.4421
  [4]: https://arxiv.org/abs/1506.02860
  [5]: https://en.wikipedia.org/wiki/Beal_conjecture