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 https://mathoverflow.net/questions/445395/asymptotic-behavior-of-a-strange-arithmetic-function