Beautiful inequality

Let's assume $a+b+c=3$ . The given expression reduces to $\sum_{cyc} \frac{a^5+a^3+14}{5-a}$ let $f(x)=\frac{x^5+x^3+14}{5-x}$ . Since $f(x)$ is a convex function, by Jensen's Inequality, we have $f(\frac{a+b+c}{3}) \leq \frac{f(a)+f(b)+f(c)}{3}$ so, we have $f(a)+f(b)+f(c) \geq 12$ . And hence the minimum value is $\boxed{12}$