Example 9. Find the integral $\int \sqrt{x^{2}+8 x+25} d x$.
Solution. This is an integral of the form (4.25). By transforming the quadratic trinomial and applying formula (4.26), we find

$$
\begin{aligned}
& \int \sqrt{x^{2}+8 x+25} d x=\int \sqrt{\left(x^{2}+8 x+16\right)-16+25} d x= \\
& =\int \sqrt{(x+4)^{2}+9} d(x+4)=\frac{x+4}{2} \sqrt{(x+4)^{2}+9}+ \\
& \quad+\frac{9}{2} \ln \left|(x+4)+\sqrt{(x+4)^{2}+9}\right|+C= \\
& =\frac{x+4}{2} \sqrt{x^{2}+8 x+25}+\frac{9}{2} \ln \left|x+4+\sqrt{x^{2}+8 x+25}\right|+C
\end{aligned}
$$