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.

First it's useful to expand
$$ab(a+b)+bc(b+c)+ca(c+a)=a^2b+b^2a+b^2c+c^2b+c^2a+a^2c$$
By Cauchy-Schwarz
$$\left(\sum_{cyc} ab\sqrt {{a\over b}(c+a)(c+b)}\right)^2\le\left( \sum_{cyc}a^2(c+b)\right)\left(\sum_{cyc}ab(c+a)\right)$$
now expand the RHS :
$$\left( ab(a+b)+bc(b+c)+ca(c+a) \right) \left(3abc+a^2b+b^2c+c^2a\right)$$
and recall that $3abc\le b^2a+c^2b+a^2c$ by AM-GM.
Hence we get
$$RHS\le \left( ab(a+b)+bc(b+c)+ca(c+a) \right)^2$$
and this is the desired inequality.
