The equation $(x-d{)}^5+x^5+(x+d{)}^5=y^n$

In this paper, we solve the equation of the title under the assumption that $\gcd (x,d)=1$ and $n\ge 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16]. Our main tools include Frey-Hellegouarch curves and associated modular forms, and an assortment of Chabauty-type techniques for determining rational points on curves of small positive genus.