Bound of an integral function

Let $$f(r):=\int_{\mathbb{R}^d}\left|\int_{\mathbb{R}^d}e^{2\pi i \langle x,y\rangle}e^{-|y|^4+r^{1/2}|y|^2}dy\right|dx,$$for $r\geq 0$.

Is it possible to uniformly bound $f$ on $\mathbb{R}_+$? i.e.$$\exists c>0,\forall r>0,f(r)\leq c$$if not, can we do it on intervals $r \in [0,U]$? That is, do you have $\sup_{r\in[0,U]}f(r)<\infty$?

Tried writing Taylor expression for cosine and exponential but this seems to complicate the problem.

Using Fourier transform properties we do have that for $r\in[0,U],h_r(y):=e^{-|y|^4+r^{1/2}|y|^2},c=\int_{\mathbb{R^d}}1/(|x|+1)^{1+d}dx$,$$\begin{align*}f(r) &\leq c\sup_{x\in\mathbb{R^d}}|(1+|x|)^{1+d}\int_{\mathbb{R}^d}e^{-2\pi i \langle x,y\rangle}e^{-|y|^4+r^{1/2}|y|^2}dy| \\&\leq c\sup_{x\in \mathbb{R}^d}\sum_{k_1+...+k_{d+1}=d+1}c_{k_1,...,k_{d+1}}|\prod_{q=1}^kx_q^{k_q}\hat{h}_r(x_1,...,x_d)| \\&\leq \sum_{k_1+...+k_{d+1}=d+1}c'_{k_1,...,k_{d+1}}\int_{\mathbb{R^d}}\partial^{k_1}_{x_1}...\partial_{x_d}^{k_d}h_r(x)dx.\end{align*}$$Would this lead to a bound on $[0,U]$?