Minimizing the Expression

$\large \sum_{cyc}{\frac{1}{a^3(b+c)}} = \sum_{cyc}{\frac{\frac{1}{a^2}}{a(b+c)}} \geq \frac{(\frac{1}{a}+\frac{1}{b} + \frac{1}{c} )^2}{2(ab+bc+ac) } \normalsize \text{ (Titu's/Cauchy Swartz)}$

$\large = \frac{ab + bc +ca}{2} \geq \boxed{\frac{3}{2}} \normalsize \text{ (AM - GM) }$

Equality holds when $a = b = c = 1$ .

If the product is constant, the sum is minimum when all of terms are equal, so a=b=c=1, then substitute by one variable