[tex]\displaystyle \int_{0}^1 \dfrac{x+1}{\sqrt{3x^2+6x+7}}\, dx = \dfrac{1}{6}\int_{0}^1 \dfrac{6x+6}{\sqrt{3x^2+6x+7}}\, dx = \\ \\ = \dfrac{1}{6}\int_{0}^1 (3x^2+6x+7)'\cdot (3x^2+6x+7)^{-\frac{1}{2}}\, dx = \\ \\ = \dfrac{1}{6}\cdot \dfrac{(3x^2+6x+7)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\Bigg|_{0}^1 =\dfrac{1}{3}(3x^2+6x+7)^{\frac{1}{2}}}\Bigg|_{0}^1 = \\ \\ = \dfrac{4}{3} - \dfrac{\sqrt 7}{3} = \boxed{\dfrac{4-\sqrt 7}{3}}[/tex]
