For $a,b,c>0.$ Prove$:$ $${\frac {ab}{ \left( a+b \right) ^{2}}}+{\frac {bc}{ \left( b+c \right) ^{2}}}+{\frac {ac}{ \left( c+a \right) ^{2}}}+\,{\frac { \left( a+b \right) \left( b+c \right) \left( c+a \right) }{16abc}}\geqslant \frac{5}{4}$$ AM-GM kills it easy, but I think it's hard to get SOS$,$ I can't!
If $c=\min\{a,b,c\},$ we obtain the following by Maple$:$
However it's ugly. So I wish another SOS.
PS: This inequality is from Nguyen Viet Hung.
There is the AM-GM proof here: https://www.facebook.com/groups/1486244404996949/permalink/2695082927446418/
So I don't need the AM-GM proof.