Answer
$\displaystyle \frac{1}{3}x^{3}+\frac{1}{2}\ln|x^{2}+9|+\frac{2}{3}\arctan\frac{x}{3}+C$
Work Step by Step
$I=\displaystyle \int\frac{x^{4}+9x^{2}+x+2}{x^{2}+9}dx$
$\displaystyle \frac{x^{4}+9x^{2}+x+2}{x^{2}+9}=\frac{x^{2}(x^{2}+9)+x+2}{x^{2}+9}=x^{2}+\frac{x+2}{x^{2}+9}$
$=x^{2}+\displaystyle \frac{\frac{1}{2}(2(x+2))}{x^{2}+9}$
$=x^{2}+\displaystyle \frac{1}{2}[\frac{2x}{x^{2}+9}]+\frac{2}{x^{2}+3^{2}}$
$\displaystyle \int\frac{x^{4}+9x^{2}+x+2}{x^{2}+9}dx=\frac{1}{3}x^{3}+\frac{1}{2}\ln|x^{2}+9|+\frac{2}{3}\arctan\frac{x}{3}+C$
