We present nonlinear shear and uniaxial extensional measurements on a series of polybutadienes with varying amounts of long-chain, random branching. Startup of steady shear experiments is used to evaluate the damping function of the melts. The damping function is found to show a trend toward decreased dependency on strain with increasing branching content. Interior chains, which are believed to be responsible for changing the damping function, are calculated to comprise less than 3 wt % of the melt. Extensional measurements are used to investigate the role of branching in strain hardening. We show that samples with increased branch contents do show larger deviations of the transient Trouton ratio from the linear viscoelastic limit of three. However, we also show that the extensional data can be fit using parameters determined solely by the shear measurements. Furthermore, we show that the changes in the damping function seen in shear have little impact on extensional behavior. The extensional behavior of the melt is found to be most affected by changes in the relaxation spectra which can result from both branching and increases in the high end of the molecular weight distribution. This statement runs contrary to the often expressed view that strain hardening behavior in extension is exclusively produced by branching.