ClearAll[inverseAbsoluteThirdOrder]

(*1/|x+r|
=(x^2+r^2+2 x.r)^(-1/2)=(1/|r|)(1+x^2/r^2+2 x.r)^(-1/2)\approx (1/|r|)(1-(x^2/r^2+2 x.r)/2+3 (x^2/r^2+2 x.r)^2/8)*)
inverseAbsoluteThirdOrder[x_,r_]:=(1/Norm[r])(1-((x.x)/(r.r)+2 x.r/(r.r))/2+3 ((x.x)/(r.r)+2 x.r/(r.r))^2/8) ;

x1={"((Subscript[x, 1])^((1)))","((Subscript[x, 2])^((1)))","((Subscript[x, 3])^((1)))"};
x2={"((Subscript[x, 1])^((2)))","((Subscript[x, 2])^((2)))","((Subscript[x, 3])^((2)))"};
r={"(Subscript[R, 1])","(Subscript[R, 2])","(Subscript[R, 3])"};

taylor =((1/Norm[r]+inverseAbsoluteThirdOrder[x1-x2,r]-inverseAbsoluteThirdOrder[x1,r]-inverseAbsoluteThirdOrder[x2,-r])) // FullSimplify

explicit = x1.x2/Norm[r]^3-3 (x1.r)(x2.r)/Norm[r]^5 // Expand

taylor - explicit // FullSimplify