When I use Mathematica to draw a bode plot, it always gives the precise version, like this:
BodePlot[(10 (s + 3))/(s (s + 2) (s^2 + s + 2))] 
How can I draw the asymptotic version, the kind where 1/(1 + jw) is represented as a -20dB straight line but not a curve?
We can draw the asymptotes using Epilog. The frequencies of interest are the max and min values and the corner frequencies.
tfm = (10 (3 + s))/(s (2 + s) (2 + s + s^2)); minF = Log10[0.001]; maxF = Log10[100]; cF1 = Log10[Sqrt[2]]; cF2 = Log10[2]; cF3 = Log10[3]; xcoords = {minF, cF1, cF2, cF3, maxF}; The magnitude at the lowest frequency.
mag0 = 20 Log10@Abs[tfm /. s -> I 0.001]; The magnitudes computed at the other frequencies of interest based on the slopes of the asymptotes. For example, between $minF$ and $cF1$ the slope is -20 db/decade.
ls = {mag0, -20 (cF1 - minF), -60 (cF2 - cF1), -80 (cF3 - cF2), -60 (maxF - cF3)}; ycoords = FoldList[Plus, ls[[1]], ls[[2 ;; -1]]]; Finally render the plot.
BodePlot[tfm, Epilog -> {{Dashed, Line[Thread[{xcoords, ycoords}]]}, {}}][[1, 1, 1]] 
The same process repeated for 10/(s+1).
tfm = 10/(s + 1); minF = Log10[0.001]; maxF = Log10[100]; cF1 = Log10[1]; xcoords = {minF, cF1, maxF}; mag0 = 20 Log10@Abs[tfm /. s -> I 0.001]; ls = {mag0, 0, -20 (maxF - cF1)}; ycoords = FoldList[Plus, ls[[1]], ls[[2 ;; -1]]]; BodePlot[tfm, Epilog -> {{Dashed, Line[Thread[{xcoords, ycoords}]]}, {}}][[1,1,1]] 