\int\frac{dx}{2x+\sqrt{3x+1}}=/t=\sqrt{3x+1}/=\int\frac{2t}{2t^{2}+3t-2}dt=\int\frac{2t}{(2t-1)(t+2)}dt=\frac{1}{5}\int\frac{2dt}{2t-1}+\frac{4}{5}\int\frac{dt}{t+2}=\frac{1}{5}ln|2t-1|+\frac{4}{5}ln|t+2|+C=\frac{1}{5}ln|2\sqrt{3x+1}-1|+\frac{4}{5}ln|\sqrt{3x+1}+2|+C