$$\frac{128 r t \, _1F_2\left(2;\frac{5}{2},\frac{7}{2};r^2 t^2\right)}{45 \pi } + \frac{2 r t+4 r t I_2(2 r t)-2 I_1(2 r t)}{r^3 t^3}$$
My computer and I found that this is the moment generating function of a distribution rooted in the geometric and statistical process of finding the mean distance between two points inside and/or on a disk of radius $r$, $r>0.$
I can factor: $$\frac{2 \left(64 r^4 t^4 \, _1F_2\left(2;\frac{5}{2},\frac{7}{2};r^2 t^2\right)+45 \pi r t+90 \pi r t I_2(2 r t)-45 \pi I_1(2 r t)\right)}{45 \pi r^3 t^3}$$
That makes it at least the product of moments so I can look for the PDF for the denominator. Question is this: What PDF corresponds to the MGF I found?
