Bounding $\int_0^T\|u\|_{L^2}^{(6-s)r/2s}\|u\|_{L^6}^{(3s-6)r/2s}$ when $u\in L^\infty(0,T; L^2) \cap L^2(0,T;L^6)$ and $\frac2r +\frac3s\geq\frac32$

Question

Suppose $u \in L^\infty(0, T; L^2) \cap L^2(0,T; L^6)$ and $$\frac2r + \frac3s = \frac32, \quad 2 \leq s \leq 6.$$ I am following a proof that states that $$\int_0^T \|u\|_{L^2}^{(6-s)r/2s} \|u\|_{L^6}^{(3s-6)r/2s}$$ is bounded when $(3s-6)r/2s \leq 2$, or equivalently, when $$\frac2r + \frac3s \geq \frac32.$$ I am having some trouble seeing why this is, I am guessing part of it has to due with working in Bochner spaces which I am very new to. How does this bound imply that the integral is finite?