Prove that for any $a, b, c>0\in \mathbb{R}^+$ the following inequality is true:↵
\begin{align*}↵
\left(\frac{a^2+b^2+c^2+b+c}{3}\right)\left(\frac{b^3}{a}+\frac{c^3}{3/2}}{\sqrt{a}}+\frac{c^{3/2}}{\sqrt{b}}+\frac{a^3}{3/2}}{\sqrt{c}}\right) \\↵
\ge a(2b-a)+b(2c-b)+c(2a-c)↵
\end{align*}
\begin{align*}↵
\left(\frac{a
\ge a(2b-a)+b(2c-b)+c(2a-c)↵
\end{align*}
