Find : $\displaystyle \int \frac{3 x+5}{\sqrt{x^{2}+2 x+4}} \mathrm{~d} x$

$$\begin{aligned} & I=\int \frac{3 x+5}{\sqrt{x^{2}+2 x+4}} d x=\frac{3}{2} \int \frac{2 x+2}{\sqrt{x^{2}+2 x+4}} d x+2 \int \frac{1}{\sqrt{x^{2}+2 x+4}} d x \\ & I=\frac{3}{2}\left(2 \sqrt{x^{2}+2 x+4}\right)+2 \int \frac{1}{\sqrt{(x+1)^{2}+\sqrt{3}^{2}}} d x \\ & I=3 \sqrt{x^{2}+2 x+4}+2 \log \left|(x+1)+\sqrt{x^{2}+2 x+4}\right|+c \end{aligned} $$