We have: $\frac{\sqrt{a^3c}}{2\sqrt{b^3a}+3bc}=\frac{a\sqrt{ac}}{b(2\sqrt{ba}+3c)}=\frac{(\sqrt{\frac{a}{b}})^2}{2\sqrt{\frac{b}{c}}+3\sqrt{\frac{a}{b}}}$
Establish similar expressions:
................................................
Set:
$(\sqrt{\frac{a}{b}};....;.....)=(x;y;z)\Rightarrow xyz=1$
By inequality Cauchy:
$\frac{x^2}{2y+3z}+\frac{2y+3z}{25}\geq \frac{2x}{5}$
Similar;
......................................
$\rightarrow P\geq \frac{1}{5}(x+y+z)\geq \frac{3}{5}$
$\rightarrow P_{min}=\frac{3}{5}$ at $a=b=c./$