\int\sqrt{1+e^{x}+e^{2x}}dx=\int\sqrt{\frac{3}{4}+(\frac{1}{2}+e^{x})^{2}}dx=/t=e^{x}/=\int\frac{\sqrt{\frac{3}{4}+(\frac{1}{2}+t)^{2}}}{t}dt=/p=\frac{1}{t}/=\int\frac{\sqrt{p^{2}+p+1}}{p^{2}}dp=/p=tgq/=\int\frac{\sqrt{tg^{2}q+1}}{tg^{2}q}dq=\int\frac{dq}{cosq}+\int\frac{cosq}{sin^{2}q}dq=ln|tgq+\sqrt{1+tg^{2}q}|-\frac{\sqrt{1+tg^2q}}{tgq}+C=ln|p+\sqrt{p^{2}+1}|-\frac{\sqrt{p^{2}+1}}{p}+C=ln|\frac{1+\sqrt{t^{2}+1}}{t}|-\sqrt{t^{2}+1}+C=ln|\frac{1+\sqrt{e^{2x}+1}}{e^{x}}|-\sqrt{e^{2x}+1}+C