Define function $$f(\alpha)=\frac{-(4-5\alpha-4\alpha^{2})+\sqrt{32\alpha^{3}+\alpha^{2}-40\alpha+16}}{(1-2\alpha)\alpha}$$ where $\alpha \in [0,\frac{1}{2}]$ I can verify that:

(1) $f(\alpha)\geq 0$ always holds when $\alpha \in [0,\frac{1}{2}]$

(2) Using L'Hôpital's rule to show $f(0)=0$ and $f(\frac{1}{2})=6$

I can plot this function via MATLAB and observes this function is increasing when $\alpha \in [0,\frac{1}{2}]$.

My goal is to prove $f(\alpha)$ is increasing in $\alpha \in [0,\frac{1}{2}]$, the geometric graph is simple but the verification of increasing is not easy. How to show it?