General formula for the integral w.r.t to Marchenko-Pastur density, of the ratio of degree $\le 2$ polynomials

Question

Question. Is there a closed-form formula (via standard objects like rational functions, radicals, special functions, special numbers like Catalan numbers, etc.) expressing integrals of rational functions w.r.t to the Marchenko-Pastur density ?

More precisely, let $p(x)$ and $q(x)$ be low-degree polynomials (say of degree at most 2) over $\mathbb R$, and let $r(x):=p(x)/q(x)$, wherever the ratio makes sense. Let $\gamma \in (0,1)$ and let $\mu_\gamma$ be the Marchenko-Pastur distribution with parameter $\gamma$ supported on $[a,b]$ where $a, b := (1 \pm \sqrt{\gamma})^2$.

Question. What is a closed-form formula (in the sense above) for $\int_a^b r(x)d\mu(x)$ in terms $\gamma$ and the coefficients of $p$ and $q$ ?

Answer 1

These integrals have a closed-form expression, but they are very lengthy if you keep the polynomials completely general. Here is the indefinite integral for linear $p(x)$ and $q(x)$, $$4 \pi d^3 {\gamma}\int\frac{a+bx}{c+dx}d\mu(x)=d \sqrt{- {\gamma}^2+2 {\gamma} (x+1)-(x-1)^2} \biggl(2 a d-b \bigl(2 c+d ( {\gamma}-x+1)\bigr)\biggr)$$ $$\qquad+2\, {\rm arctan}\,\left(\frac{ {\gamma}-x+1}{\sqrt{- {\gamma}^2+2 {\gamma} (x+1)-(x-1)^2}}\right) \bigl(-a d (c+d {\gamma}+d)+b d {\gamma} (c-2 d)+b c (c+d)\bigr)$$ $$\qquad +2 \sqrt{c^2+2 c d ( {\gamma}+1)+d^2 ( {\gamma}-1)^2} (b c-a d)\, {\rm arctan}\,\left(\frac{c (- {\gamma}+x-1)+d ( {\gamma}+1) x-d ( {\gamma}-1)^2}{\sqrt{- {\gamma}^2+2 {\gamma} (x+1)-(x-1)^2} \sqrt{c^2+2 c d ( {\gamma}+1)+d^2 ( {\gamma}-1)^2}}\right).$$