\int\sqrt{\varphi^2+\rho^2x^2}dx=\varphi\int\sqrt{1+\left(\frac{\rho}{\varphi}x\right)^2}dx=\frac{\varphi^2}{\rho}\int\sqrt{1+\left(\frac{\rho}{\varphi}x\right)^2}d\left(\frac{\rho}{\varphi}x\right)