  Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-\Delta
u-\frac{\gamma}{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in
$\mathbb{R}^N\setminus \{0\},$ where
$0<\gamma<\gamma_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and
$2^*(s)=\frac{2(N-s)}{(N-2)}.$ We prove a stability estimate for the
corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991).
Also, we obtain a Struwe-type decomposition (1984) for the corresponding
Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble,
namely $\operatorname{dist}(u,\mathcal{M})\lesssim \Gamma(u)$ in the spirit of
Ciraolo-Figalli-Maggi (2017).
