ClearAll["Global`*"]
assume = {r > 0, Mh > 0, 0 < aspin < 1, 0 < bhrms < 1, md > 0,
   G > 0, \[Sigma]b > 0, c > 0};
r0 = 2 r/((G Mh)/c^2);
y = Sqrt[r0];
yms = Sqrt[bhrms];
spinacos = ArcCos[aspin];
y1 = 2*Cos@(1/3*(spinacos - \[Pi]));
y2 = 2*Cos@(1/3*(spinacos + \[Pi]));
y3 = -2*Cos@(1/3*spinacos);

part3 = 3*((y3 - aspin)^2)*Log@((y - y3)/(yms - y3));
part3 = part3/(y*y3*(y3 - y1)*(y3 - y2));
part3 = FullSimplify[part3, assume];
part2 = 3*((y2 - aspin)^2)*Log@((y - y2)/(yms - y2));
part2 = part2/(y*y2*(y2 - y1)*(y2 - y3));
part2 = FullSimplify[part2, assume];
part1 = 3*((y1 - aspin)^2)*Log@((y - y1)/(yms - y1));
part1 = part1/(y*y1*(y1 - y2)*(y1 - y3));
part1 = FullSimplify[part1, assume];
cc = 1 - yms/y - (3*aspin/(2*y))*Log@(y/yms) - part1 - part2 - part3;
cc = FullSimplify[cc, assume];
bb = 1 - 3/r0 + 2*aspin/(r0^(3/2));
bb = FullSimplify[bb, assume];
FullSimplify[((3 G Mh md)/(8 \[Pi] \[Sigma]b r^3) cc/bb)^(1/4),
 Assumptions -> assume]