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, $$T(n)=O(n(log_{5/4}(log_2 n))^2)$$