Are you getting carted three times or are you just not doing enough damage and you run out of time?
If it's the first, I would set out on the quest with the mindset that I'll fight the monster earnestly, but mostly I'll practice evading her attacks, and watching what she does between attacks so I can learn when is a good time to attack her, and what position would be best. Pay attention to where you were in relation to her when she gets big hits on you or combo's you, so you can avoid that position in the future. Don't worry about winning or losing, this time it's just about learning.
When I fight the Rathian, I like to get to the side of her head to hit her head while she fireballs.
When she does the flip over to poison you thing, I like to be a little to the side of her tail to get hits on her tail.
Take note of your habits. I have a really bad habit: when I get in a good position to get alot of hits at once, I tend to continue a little too long. Or I get overexcited and I button-mash, making my attack last longer than I want it to. Either way, I get hit when I do this. So I have to discipline myself. If I think I can get 4 hits in, I'll force myself to stop at 3 hits, evade, then find another postion.
If you're not getting regularly carted and you just don't seem to have enough time, then I would start using traps. And some bombs. It's not always easy to get her into a pitfall trap, but try it. Lay the bombs at her head. Or just hit her head as much as you can while she flails in the trap. If you can get her in the trap while she's drooling, it will keep her there longer. I like to use shock traps to get hits on her tail when I get frustrated trying to get that tail off.
I don't remember what armors are available at that level, but if I remember correctly, the Barroth gives you the best defense? I never owned gobul armor, because I hated gobul for a long time.) I wish I could remember what skills were available, but here are the skills I like:
Perception: I like to know immediately when I can trap and tranq the monster in case I run out of time to get a kill. (Remember that if you use lots of traps, they last fewer seconds each time. If I've used alot of traps, and I want to trap and tranq, I try to pre-tranq her so hopefully I will only have to throw one. Or if it goes really well, none.)
Speed setuip/Trapmaster: gives me alot more control over setting traps without having to take damage as often, lets me get 2 Lg. Barrel Bombs down and 1 small before the trap expires. All your trap mixes are 100 percent. I hate getting garbage when I'm relying on that trap mix to work.
Rathian + switch axe is all about the timing and positioning. Try to keep from getting in front of her too much (fireballs) or directly under her (poison flip, tripping you with her feet). I like to stay to the side as much as possible.
Once you can access the volcano, I like the ingot armor for the Rathian. Not much for Fire res, but I really like the Wind res for those big bird things.
As far as what Chips said about upswinging and axe mode, well, I was going to suggest the opposite. I don't like being under her when she's flying, I just get poisoned or hit. I try to stay to the side and hit her tail. Or flashbomb her out of the air if she's not raging. And sword mode for me seems more effective, with faster, smoother swings. Which goes to show you that with any advice you get from anyone, your mileage may vary. If you can try to hit her out of the air without getting poisoned, go for it! We all find our own unique approaches that work best for us.
(Btw, my SA of choice for the Rathian was the Rough Edge, used mostly in sword mode. Plan to sharpen it alot, though, if you decide to try it. For me, it just worked alot better than the other blade.)