I'm going to assume you know to use integration by parts here.
Once you establish the x^6 term in the 2nd part of the integral, split it into x and x^5, and then in the x^5 term, expand it into -([1-x^5]-1). Now expand with the (1-x^5)^n-1 term. Group I_n on the LHS.
I don't want to give too much away because this is a relatively common technique used in these reduction formulae.