Hamilton's principle with semiholonomic constraints in Goldstein

I am studying from Goldstein’s Classical Mechanics, 3rd edition. In section 2.4, he discussed Hamiltion’s principle with semiholonomic constraints. The constraints can be written in the form $f_alpha(q_1,…,q_n;dot{q_1},…,dot{q_n};t)=0$ where $alpha=1,…,m$. Using variational priciple, we get

$$deltaint_{t_1}^{t_2}left(L+sum_{alpha=1}^m mu_alpha f_alpharight)mathrm dt=0 tag{2.26}$$

where $mu_alpha=mu_alpha(t)$.

But how can he get the formula

$$frac{d}{dt}frac{partial L}{partialdot{q_k}}-frac{partial L}{partial q_k}=-sum_{alpha=1}^m mu_alpha frac{partial f_alpha}{partialdot{q_k}} tag{2.27}$$

for $k=1,…,n$ from the previous formula?

When I go through the steps as in section 2.3, I get
$$frac{dI}{dbeta}=int_{t_1}^{t_2}sum_{k=1}^nleft(frac{partial L}{partial q_k}-frac{d}{dt}frac{partial L}{partialdot{q_k}}+sum_{alpha=1}^m mu_alphaleft(frac{partial f_alpha}{partial q_k}-frac{d}{dt}frac{partial f_alpha}{partialdot{q_k}}right)right)frac{partial q_k}{partialbeta}mathrm dt$$
where $beta$ denotes the parameter of small change of path:
begin{align}
q_1(t,beta)&=q_1(t,0)+betaeta_1(t)
q_2(t,beta)&=q_2(t,0)+betaeta_2(t)
& ,,vdots
end{align}
Using the same argument as in the part of holonomic constraint in section 2.4, I get
$$frac{partial L}{partial q_k}-frac{d}{dt}frac{partial L}{partialdot{q_k}}+sum_{alpha=1}^m mu_alpha left(frac{partial f_alpha}{partial q_k}-frac{d}{dt}frac{partial f_alpha}{partialdot{q_k}}right)=0$$
for $k=1,…,n$.

What am I missing?