One has
$$\sum_{n \leq x}\frac{n}{\operatorname{rad}n} \sim x \sqrt{\frac{2}{\log x \log \log x}} \exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty),$$
according to user "Ofir Gorodetsky"'s detailed answer to a similar question to yours at https://mathoverflow.net/questions/445395/asymptotic-behavior-of-a-strange-arithmetic-function

The exp term grows faster than any power of $\log x$, so dividing by $\sqrt{\log x \log \log x}$ does nothing to change that.