Cauchy-schwarz $$(\sum_{cyc}a\sqrt{8b^2+c^2})^2 \leq \sum_{cyc}a(51a+100b+2c)\sum_{cyc}\frac{a(8b^2+c^2)}{51a+100b+2c}=51(a+b+c)^2\sum_{cyc} \frac{a(8b^2+c^2)}{51a+100b+2c}$$ It suffices to prove that
$$51\sum_{cyc}\frac{a(8b^2+c^2)}{51a+100b+2c} \leq (a+b+c)^2$$
we can easily check this inequality.
