Hello, everyone. I want to calculate the expectation shown in the following formula, where $X$ follows a standard $d$-dimensional multi-variable normal distribution as $X\sim\mathbb{N}(\mathbf{0},\mathbb{I}_{d\times d})$ $$ E_X\left[\frac{X^\top\mathbf{A}XX^\top\mathbf{B}X}{\|X\|^2}\right] $$ where $\|X\|^2=X^\top X$, and $\mathbf{A},\mathbf{B}$ are both real symmetric matrices.

The expectation of the numerator is straightforward and there was result for it. However, the denominator seems to make the problem more difficult. Are there any simple approaches to deal with this problem? Thank you very much!