In my research I've recently stumbled upon integrals of the form: $$ \int_{S^2} \frac{x^T B x}{x^T A x} $$ where $A$ is a symmetric positive-definite $3\times 3$ matrix and $B$ is just a symmetric $3\times 3$ matrix.

I was wondering whether anything was known about these types of integrals?

In my research I've recently stumbled upon integrals of the form: $$ \int_{S^2} \frac{x^T B x}{x^T A x} \,dx $$ where $A$ is a symmetric positive-definite $3\times 3$ matrix and $B$ is just a symmetric $3\times 3$ matrix.

I was wondering whether anything was known about these types of integrals?