$\int\limits \frac{3x+5}{2x^2+x+10}dx=\int\limits [\frac{3(4x+1)}{4(2x^2+x+10}+\frac{17}{4(2x^2+x+10)}]dx$ $=\int\limits [\frac{3(4x+1)}{4(2x^2+x+10)}+\frac{17}{8[(x+\frac{1}{4})^2+\frac{79}{16}]}]dx$
$=\frac{3}{4}ln(2x^2+x+10)+\frac{17}{2\sqrt{79}}artan\frac{4x+1}{\sqrt{79}}+C$