$\sqrt {{x^2} + 4x + 6}$

Let $I = \int {\sqrt {{x^2} + 4x + 6} } dx = \int {\sqrt {{x^2} + 4x + 4 + 2} } dx$

$= \int {\sqrt {{{\left( {x + 2} \right)}^2} + {{\left( {\sqrt 2 } \right)}^2}} dx}$

$= \cfrac{{x + 2}}{2}\sqrt {{{\left( {x + 2} \right)}^2} + 2} + \cfrac{2}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{{\left( {x + 2} \right)}^2} + 2} } \right| + C$

$= \cfrac{{x + 2}}{2}\sqrt {{x^2} + 4x + 6} + \log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 6} } \right| + C$