Invariant to inversions 2 
Let $a,b,c,d,e,f>0$ satisfying
  $a+b+c+d+e+f=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}$
  . Prove
  $$ab+bc+cd+da+ac+bd+ae+be+ce+de+af+bf+cf+df+ef+10\sqrt{abcdef}\geq25$$

Attempt: Prove that if $k>10$ then the inequality
$$ab+bc+cd+da+ac+bd+ae+be+ce+de+af+bf+cf+df+ef+k\sqrt{abcdef}\geq15+k$$is not always true.
Also, I tried 
$$a+b+c+d+e+f\geq\frac{36}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}}=\frac{36}{a+b+c+f+e+f}.$$
Also, $(a+b+c+d+e+f)^2$ appears $ab+ac+ad+...+ec+ed+ef.$

Let $a+b+c+d+e+f= const$ and $a^2+b^2+c^2+d^2+e^2+f^2=const.$
Thus, by the Vasc's EV Method : https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf (corollary 1.8 (b)) 
it's enough to prove our inequality for equality case of five variables.
Now, let $b=c=d=e=f$ and $a=xb.$
Thus, the condition gives $$a+5b=\frac{1}{a}+\frac{5}{b}$$ and we need to p-rove that
$$5ab+10b^2+10\sqrt{ab^5}\geq25$$ or
$$x+2+2\sqrt{\frac{5x+1}{x+5}}\geq\frac{5x(x+5)}{5x+1}$$ or
$$\sqrt{\frac{5x+1}{x+5}}\geq\frac{7x-1}{5x+1}$$ and the rest is smooth.
