By AM-GM, $5x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}}\geq 6x^2$, hence done.

Emirhan wrote: If $x$ and $y$ are positive reals, prove that $$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$ hello, it is $$\frac{(x-y)(x^5-y^5)}{xy}\geq 0$$Sonnhard.

Alternatively, use the rearrangement inequality on the sequences $\{ x^2\sqrt[4]{\frac{x}{y}},y^2\sqrt[4]{\frac{y}{x}} \}$ and $\{\sqrt[4]{\frac{x}{y}},\sqrt[4]{\frac{y}{x}} \}$ .