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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

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 $a=b=c./$