I've tried to expand the recursion:
$$T(n) = T(n^{\frac{1}{2}}) + 5n = T(n^{\frac{1}{4}}) + 5(n^{\frac{1}{2}} + n) = T(n^{\frac{1}{8}}) + 5(n^{\frac{1}{4}} + n^{\frac{1}{2}} + n)$$
We have a total of $\lfloor\log_2(n)\rfloor + 1$ elements in the recursion. So we can easily find an upper bound: $$T(n) = O(n\cdot \log_2(n))$$
The thing is, I'm not sure it's a tight upper bound. More over, I haven't been able to reach a tight lower bound from the recursion. I've tried to, but I didn't get to the same assymptotic bound I've found above.
Any help would be appreciated.