Consider that $\beta$, $\alpha$ and $F_\beta$ are $N\times1$ vectors and $M$ is an $N\times N$ matrix. $\beta$ and $\alpha$ are constants and $F_\beta$ and $M$ are functions of $x$.
Now consider the expression
$$s_1=\int_0^L{(\alpha^TM\alpha) dx}.$$
This can easily be rewritten as, $$s_1=\alpha^T(\int_0^L{M dx})\alpha.$$
However, how does one rewrite: $$s_2=\int_0^L{(\beta^TF_\beta)(\alpha^TM\alpha) dx}.$$ in a similar way? That is, something like $$s_2=\left[\beta^T\left(\int_0^L{F_\beta dx}\right)\right]\left[\alpha^T\left(\int_0^L{Mdx}\right)\alpha\right],$$
except that the form above is not correct.