One has \begin{align*} \sum_{n \leq x}\frac{n}{\operatorname{rad}n} & = (1+o(1)) \, 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), \\ & = x\exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty) \\ & \neq O(x\, (\log x)^A) \ (x\to \infty) \end{align*}
for any $A \in \mathbb{R}$, where the first estimate holds according to user "Ofir Gorodetsky"'s detailed answer to a similar question to yours at Asymptotic behavior of a "strange" arithmetic function.
