http://i42.tinypic.com/if3614.jpgI found where it was increasing and decreasing I think. At first I added all the time together from where all the increasing parts are, but then I didn't think that was right. So then I added the parts where it was only going to be concave up, where the positive slopes were, but I'm not sure.
you're making this way too difficult when it is actually very simple.
P'(x) is the velocity shown in the graph.
the ant moves upward when P'(x) > 0 (when the graph is above the x-axis)
the time intervals where P'(x) > 0 would be ...
0 to 3 = 3
3.5 to 6 = 2.5
8 to 13 = 5
15 to 18.5 = 3.5
total = 14 minutes