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 $ngeq 2$. 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.