(p+(pq^2)^1/3+(p^2q)1/3)/p-q*(1-q1/3/p1/3)
Assuming you meant
(p+(pq^2)^1/3+(p^2q)^1/3)/p
that term is
1 + (q/p)^1/3 + (q/p)^2/3
If you let u = (q/p)^1/3 then you just have
(1+u+u^2)-q(1-u)
But q = pu^3
Not sure just where you want to go with this. Your typos make things a bit murky.