You can simplify the integrand:
[tex]\dfrac{3x^2+4x+1}{2x}=\dfrac{3x}2+2+\dfrac1{2x}[/tex]
Then integrate term-by-term:
[tex]\displaystyle\int\frac{3x^2+4x+1}{2x}\,\mathrm dx=\frac32\int x\,\mathrm dx+2\int\mathrm dx+\frac12\int\frac{\mathrm dx}x[/tex]
[tex]=\dfrac{3x^2}4+2x+\dfrac12\ln|x|+C[/tex]
