A trapped dipole for 40m and 80m offers the advantage of being somewhat shorter than a full size single band 80m resonant dipole plus it offers 40m as a resonant band plus the possibility of working on 20m, 15m and 10m. There are several designs available on the web for this type of aerial so Google W3DZZ. One of the most comprehensive sets of instructions is by Len Paget G0ONX. Fine out more here:
This would be my choice if I had the space, though since a dipole is a balanced aerial it would make more sense to use balanced twin feeder (as in the Spectrum Communications implementation of this design) rather than coaxial cable which is an un-balanced and more lossy feeder.
Here is a variation on the W3DZZ antenna by the Maidstone Amateur Radio Society that adds a dedicated 10 meter (28MHz) resonant element as a 'fan'.
W3DZZ Dipole Aerial design by the Maidstone Amateur Radio Society