Using David Trimas' integral but with slightly simpler integration limits Mathematica 14.3 (Windows 11) results in the following:

pdf2 = Integrate[8 Sqrt[(1 - x^2) (1 - (d + x)^2)]/\[Pi]^2, {x, -1, 1 - d}, 
  Assumptions -> 0 < d < 2]
(* (4 z ((4 + d^2) EllipticE[1 - 4/d^2] - 8 EllipticK[1 - 4/d^2]))/(3 \[Pi]^2) *)

Note that I used d rather than z as in David Trimas' example. I only get the input repeated back using z and even using y. I have quit and restarted the kernel, closed and reopened Mathematica and still the same issue.

Here is a "plot" of the two implementations:

pdf[z_] := If[0 < z < 2, 8/\[Pi]^2 NIntegrate[Sqrt[(1 - x^2) (1 - (x - z)^2)], {x, Max[z - 1, -1], Min[z + 1, 1]}], 0]
Plot[{pdf2, pdf[d]}, {d, 0, 2}, PlotStyle -> {Thickness[0.02], {Thickness[0.005], Yellow}}]

A numerical method to calculate the PDF:

pdf[z_] := 
 If[0 < z < 2, 
  8/π^2 NIntegrate[ 
    Sqrt[(1 - x^2) (1 - (x - z)^2)], {x, Max[z - 1, -1], Min[z + 1, 1]}],
   0]

Maybe there's some kind of trig substitution that would make $$ \frac{8}{\pi^2}\int _{\max (-1,z-1)}^{\min (1,z+1)}\sqrt{\left(1-x^2\right) \left(1-(x-z)^2\right)}dx$$ solvable?