The solution of the differential equation $(y+x\sqrt{xy}(x+y\left)\right)dx\text{\hspace{0.17em}}+\left(y\sqrt{xy}\right(x+y)-x)dy=0,$ is 

The given equation can be written as

$\Rightarrow (ydx-xdy)+x\sqrt{xy}(x+y)dx+y\sqrt{xy}(x+y)dy=0$

$\Rightarrow (ydx-xdy)+(x+y)\sqrt{xy}(xdx+ydy)=0$

$\Rightarrow \frac{ydx-xdy}{y^2}+(\frac{x}{y}+1).\sqrt{\frac{x}{y}}d\left(\frac{x^2+y^2}{2}\right)=0$

$\Rightarrow d\left(\frac{x^2+y^2}{2}\right)+\frac{d\left(\frac{x}{y}\right)}{(\frac{x}{y}+1).\sqrt{\frac{x}{y}}}=0$

$d\left(\frac{x^2+y^2}{2}\right)+2d\left({\tan }^{-1}\left(\sqrt{\frac{x}{y}}\right)\right)=0$

Integrating both the sides, we get

$\Rightarrow \frac{x^2+y^2}{2}+2{\tan }^{-1}\sqrt{\frac{x}{y}}=c$