Try to code it neatly, and first do the substitution to make it a lot easier. In addition, change the substitutions to exact fraction:
func=(-((4alpha b^3)/(2 b+mu)^2)+(64 b^5 (6 b+4 b^3-16 b^5+(3 Sqrt[1/b^2] b Pi)/Sqrt[1-b^2]-(6 ArcSin[b])/Sqrt[1-b^2]))/((96 b^5-96 b^7)(Pi)))/(1+b^2);
func=func/.{alpha->1/20,mu->1/100}//Simplify
NMinimize[{func,0<=b<1},{b},AccuracyGoal->40,WorkingPrecision->40]
