Integrate[((x^(a/2))/(((b+a x))^(((a+b))/2))),{x,0,∞},Assumptions->Element[a,Reals]&&a>0&&Element[b,Reals]&&b>2]

Integrate[((x^((a/2)+1))/(((b+a x))^(((a+b))/2))),{x,0,∞},Assumptions->Element[a,Reals]&&a>0&&Element[b,Reals]&&b>4]

Compute the variance using these expressions

Simplify[ (b^2)(((a+2))/(a (b-2) (b-4)))-((((b/(b-2))))^2)]