$\begin{align}  & \int{\frac{x+1}{{{x}^{2}}+5x+6}dx=\int{\frac{x+1}{(x+2)(x+3)}dx}} \\ & \text{Resolving }\frac{x+1}{(x+2)(x+3)}\text{ into partial fraction} \\ & \frac{x+1}{(x+2)(x+3)}=\frac{A}{x+2}+\frac{B}{x+3} \\ & x+1=A(x+3)+B(x+2) \\ & \text{Set }x=-2 \\ & A=-1 \\ & \text{set }x=-3 \\ & B=2 \\ & \int{\frac{x+1}{{{x}^{2}}+5x+6}dx}=\int{\frac{-1}{x+2}dx+\int{\frac{2}{x+3}}dx} \\ & \int{\frac{x+1}{{{x}^{2}}+5x+6}dx}=-\ln (x+2)+2\ln (x+3)+C \\ & \int{\frac{x+1}{{{x}^{2}}+5x+6}dx}=\ln \frac{{{(x+3)}^{2}}}{(x+2)}+C \\\end{align}$