Let $a,b,c > 0$ such that $a^2+b^2+c^2=3$. Prove the following inequality $$2\sum_{cyc}a^4+\sum_{cyc} ab(a^2+b^2)+8abc\sum_{cyc} a\ge 3\sum_{cyc} a^3b^3+6abc\sum _{cyc}a^2b+3abc\sum_{cyc}ab^2.$$ Maybe this paper can be useful: On the cyclic homogeneous polynomial inequalities of degree four.
The inequality arises when I solve $$\dfrac{a}{b+c^3}+\dfrac{b}{c+a^3}+\dfrac{c}{a+b^3}\ge\dfrac{3}{2},$$ using Cauchy-Schwarz's inequality $$\dfrac{a}{b+c^3}+\dfrac{b}{c+a^3}+\dfrac{c}{a+b^3}\ge\dfrac{(a^2+b^2+c^2+ab+bc+ac)^2} {a(b+c^3)(a+b)^2+b(c+a^3)(b+c)^2+c(a+b^3)(c+a)^2}$$ so that $$\Longrightarrow 2\sum_{cyc}a^4+\sum ab(a^2+b^2)+8abc\sum a\ge 3\sum a^3b^3+6abc\sum a^2b+3abc\sum ab^2.$$
