Given a $d$-dimensional standard Gaussian variable $\varepsilon$, a variable $X$ independent of $\varepsilon$ and a continuous function $F : \mathbb{R}^d \to \mathbb{R}^d$ , I am trying to compute $$\mathbb{E}[ \mathrm{exp} \, (\langle F(X), \epsilon \rangle + \tfrac{1}{4} \Vert \varepsilon \Vert^2) | X]$$
which "amounts" to computing
$$\mathbb{E}[ \mathrm{exp} \,(\Vert F(X) + \tfrac{1}{2} \varepsilon \Vert^2) | X]$$ as the first term if we develop the square is $X$-measurable. But I don't really know where to go further, as the exponetial prevents me from splitting the sum of the scalar product.
Edit: After discussion, it is related to the MGF of a non central chi-square distribution: for a fixed $x$, $N = \Vert 2 F(x) + \varepsilon \Vert^2$ follows a chi-square distribution with $d$ degrees of freedom and $\lambda = 4 \Vert F(x) \Vert^2 $. It remains to apply the formula for the MGF of a non central chi-square, with $t = 1/4$: $$\mathbb{E}[e^{t N}] = \frac{e^{\lambda /2}}{2^{d/2}}$$