Let $a,b,c$ be positive reals. Prove that $ab(a+b)+bc(b+c)+ca(c+a)\geq \sum_{cyc} ab\sqrt {{a\over b}(c+a)(c+b)}$ 
Let $a,b,c$ be positive reals. Prove that
  $$ab(a+b)+bc(b+c)+ca(c+a)\geq \sum_{cyc} ab\sqrt {{a\over b}(c+a)(c+b)}$$

I have no clue and I am not good at inequalities. Please help.

Denote $A = a^2b+b^2c+c^2a,B=ab^2+bc^2+ca^2$, we have $LHS=A+B$.
By AM-GM inequality $$ab\sqrt{{a\over b}(c+a)(c+b)} = \sqrt{a^2(b+c)\cdot ab(a+c)} \le \frac{a^2(b+c) + ab(a+c)}{2},$$
Thus 
$$RHS = \sum ab\sqrt{{a\over b}(c+a)(c+b)} \le \frac{1}{2}\sum [a^2(b+c) + ab(a+c)]=\frac{1}{2}(2A+B+3abc)\quad (1)$$
By AM-GM inequality again: $$B=ab^2+bc^2+ca^2 \ge 3\sqrt[3]{ab^2\cdot bc^2\cdot ca^2}=3abc\quad (2)$$
From $(1)$ and $(2)$ we have $$RHS \le \frac{1}{2}(2A+B+3abc) \le \frac{1}{2}(2A+B+B) = A+B = LHS,$$
QED.
