\int \frac{ \sqrt{x+\sqrt{x^2+y^2}}}{ \sqrt{x^2+y^2}} \, dy=\left|\sqrt{x^2+y^2}\right.=t, \frac{y \text{dy}}{\sqrt{x^2+y^2}}=\text{dt},\frac{ \text{dy}}{\sqrt{x^2+y^2}} =\frac{\text{dt}}{y}=\left.\frac{\text{dt}}{\sqrt{t^2-x^2}}\right|=\int \frac{ \sqrt{x+t}}{ \sqrt{t^2-x^2}} \, dt=\int \frac{ 1}{ \sqrt{t-x}} \, dt=2 \sqrt{t-x}=2 \sqrt{\sqrt{x^2+y^2}-x}