How to calculate one Cauchy type determinant As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i+y _j)}.$$
Is there something known about the following generalized Cauchy determinant?
$$\det \left(\frac{A_i+B_j}{x _i+y _j}\right) _{1\le i,j \le n}.$$
More specially, how does it go for 
$$\det \left(\frac{A_i+A_j}{x _i+x _j}\right) _{1\le i,j \le n}.$$
A simple case is for $x_i=i$. 
I wonder if there are some references about them. Thank you.
 A: See (3.1) of Okada, S. "Generalizations of Cauchy’s Determinant Identity
and Schur’s Pfaffian Identity"
A: A determinant of this type is discussed in these two publications on the two-dimensional square lattice Ising model on the rectangle:
Hucht, Alfred, The square lattice Ising model on the rectangle. I: Finite systems, J. Phys. A, Math. Theor. 50, No. 6, Article ID 065201, 23 p. (2017). ZBL1357.81154.
https://arxiv.org/abs/1609.01963 
Hucht, Alfred, The square lattice Ising model on the rectangle. II: Finite-size scaling limit, J. Phys. A, Math. Theor. 50, No. 26, Article ID 265205, 23 p. (2017). ZBL1369.82005.
https://arxiv.org/abs/1701.08722 
A: Not an answer, but: experiment seems to show no nice pattern for the numerator of the determinant, but the denominator seems to be the product of all the $x$ variables and all the squares of the sums of pairs of $x$-es.
A: As already noted by others, it does not seem feasible a general formula shapes up here. Having said that, the special case $A_j=Ay_j$ and $B_i=Bx_i$ finds a closed form involving (unsurprisingly) the Vandermonde determinant; see page 2 of the paper

Tewodros Amdeberhan and Doron Zeilberger, "Trivializing" generalizations of some Izergin-Korepin-type determinants, Discrete Mathematics and Theoretical Computer Science, DMTCS, 2007, 9 (1), pp.203–206, hal-00966504 (pdf)

To wit,
$$
\det\left(\frac{Ay_j+Bx_i}{y_j+x_i}\right)_{i,j}^{1,n}
=(A-B)^{n-1}\frac{(A\pmb{y}+(-1)^{n-1}B\pmb{x})\cdot\prod_{i<j}(x_i-x_j)(y_i-y_j)}{2\prod_{i,j}^{1,n}(x_i+y_j)};
$$
where $\pmb{x}=\prod_{i=1}^nx_i$ and $\pmb{y}=\prod_{j=1}^ny_j$.
