Simple integrals for plane current distributions.

$Assumptions = z ≠ 0 && z ∈ Reals && R > 0;
Integrate[ r (z^2 + r^2)^(-3/2), r]
Integrate[ r (z^2 + r^2)^(-3/2), {r,0,Infinity}]
(*Integrate[ r (z^2 + r^2)^(-3/2), {r,0,R}]*)

$Assumptions = r > 0;
Integrate[  (z^2 + r^2)^(-3/2), {z, -Infinity, Infinity}]