Just a suggestion: It would take less time to capture another bluebird than to fix this image, because it needs a lot more work
than just the shadow. That bright, vertical band in the background is a deal-breaker and needs to be cloned out. So is the branch
on the left that merges with the bird's bill.
As for the shadow, if you're a novice at retouching, this won't be of much help, but others my want to play with it. My approach
was first a hue/sat adj layer in screen mode with "colorize" ticked. Adjust hue and saturation. Black mask on that and paint with
white on the shadow, blending it at the bright edges. Above that, a blank layer in soft light mode. Paint with white or light blue to
brighten further. If I were playing for keeps with a large image, I would do another layer or two like that and use some lighter and darker
tones to add some texture back into the shadow area.
If it were my picture, I'd toss this one and go for another shot. Fixing this one might be good practice, but I think it has too many problems
to make it worth trying to salvage for print.