algorithms – Solve recurrence relation $T(n)=n^{1/5}T(n^{4/5})+5n/4$


I am trying to solve this recurrence relation – $T(n)=n^{1/5}T(n^{4/5})+5n/4$. I can’t use the master’s method and the recursion tree method because of that $n^{1/5}$ term.

We can write $$frac{T(n)}n=frac{T(n^{4/5})}{n^{4/5}}+frac54$$

Now we can change variable $S(n)=frac{T(n)}{n}$. So, we get $$S(n)=S(n^{4/5})+frac{5}{4}$$

Then I used the recurrence tree method by taking $n=2^{(frac{5}{4})^k}$. I got $S(n)=O((log_{5/4}(log_2 n))^2)$. So, $$T(n)=O(n(log_{5/4}(log_2 n))^2)$$

I don’t know, the answer doesn’t look right. Can anyone please help me out? If anyone knows the final answer please let me know, so that I can check.

Thanks!