Expectation of cubic form (covariance between a quadratic form and a dot product)

Question

Consider a $n$-dimensional random vector $X$ with mean $\mathbb{E}X=\mathbf{0}$ and covariance matrix $\Sigma$. Given a $n\times n$ matrix $A$ and a $n$-dimensional vector $b$, is there any explicit expression for the following quantity $$ \mathrm{Cov}(X^T A X, b^T X)=\mathbb{E}[ (X^TAX) (b^TX)] \text{ ?} $$