Pathfinder =/= DnD.
That said the bow is illegal. It requires a +1 enhancement bonus making it a +4 weapon. The bonus by masterwork is insufficient as the enhancement bonus in creating magic weapons says the bonus applies to both attack and damage but masterwork only applies to attack.
Now, to your question! :)
Vital Strike is counter to the other two. VS works on a single attack. Many Shot says you get 2 shots on the first attack of a full attack and speed works on full attack. So, Many Shot and Speed will work together but neither apply with Vital Strike.
Ninjas...
I am playing an Ranger (Bows) Currently level 6. I just got a Masterwork Speed Longbow(+3), and i have Many Shot as well as Vital Strike.
I was just wondering about damage calculating with these 2 feats?
Since I don't know what your dexterity score is, you'll have to adjust this. Also I assume you're medium. If you're playing a small character, adjust accordingly (both for weapon size and bonuses to hit). I also assume you have a +3 Speed Longbow which would cost 72,375 gp and is WAY better than a level 6 character should normally have.
Ranger 6 = BAB+6
Longbow enhancement bonus = +3
General attack roll formula: BAB + Dex + Enhancement + Misc
Attack action (Standard action) one attack only: d20+9 deals 2d8 piercing damage.
That's an attack roll of +9 (BAB6 + Enhancement3) and you get an extra d8 from vital strike.
Full-Attack (Full-round action)
Attack #1: d20+9 deals two sets of 1d8 piercing damage. (Static bonuses like favored enemy will apply to each one)
Attack #2 (Iterative attack from high BAB): d20+4 deals 1d8 piercing damage.
Attack #3 (Speed weapon/Haste): d20+9 deals 1d8 piercing damage.
Attacks #1 and #3 are at full BAB, so it's just like the first example. Attack #2 is at BAB-5 (because that's how iterative attacks work) so it's BAB 6-5 = 1, plus enhancement (plus dex, etc.).
You can make attack #3 at any time, before #1, or in between them. The only thing that matters as far as order of attacks is that you must make #1 before #2 because those attacks are granted by high BAB. This isn't really important.
So taking another route, would you by chance mean that you just received a +3 Speed Longbow? +3 being the enhancement bonus. Or did you mean that the weapon at hand is a Composite Longbow with a +3 Strength rating?
As it has been said, normally a weapon is required to have some type of enhancement bonus before you place a special ability on it like Speed.
This information will help us calculate things out, but as it has been said, Vital Strike requires a standard action and thusly will not come into play when you use the Speed property of a weapon and/or Manyshot. However, Manyswhot and the Speed property will both come into effect when you take a full attack action.
Once we understand a bit more about the bow you have, we can help you calculate out these different attacks.
Long Bow with Speed Enchant that i got in our last boss encounter (yes i know this is abit good for a level 7 PC. When my DM rolled loot his face went funny because he rolled really high apparently as well as i am now at -30000g in the party loot)
Is it a +1 Speed Longbow? Or a +3 Speed Longbow?
Or is it a custom magic item your GM created?
He said Its a Longbow with the Speed enchant
Magic Weapons: "A weapon with a special ability must also have at least a +1 enhancement bonus."
So, assuming it's a custom magic item, that means:
Ranger 6 = BAB+6
Dexterity 20 = +5
Longbow masterwork enhancement bonus = +1
General attack roll formula: BAB + Dex + Enhancement + Misc
Attack action (Standard action) one attack only: d20+12 deals 2d8 piercing damage.
That's an attack roll of +12 (BAB 6 + Dex 5 + Enhancement 1) and you get an extra d8 from vital strike.
Full-Attack (Full-round action)
Attack #1: d20+12 deals two sets of 1d8 piercing damage. (BAB 6 + Dex 5 + Enhancement 1)
Attack #2 (Iterative attack from high BAB): d20+7 deals 1d8 piercing damage. (BAB 6 Iterative -5 + Dex 5 + Enhancement 1)
Attack #3 (Speed weapon/Haste): d20+12 deals 1d8 piercing damage. (BAB 6 + Dex 5 + Enhancement 1)
That's three attack rolls, and 4 arrows if all of them hit.
If your longbow was +1 then the enhancement bonus to attack would not stack with the bonus from masterwork, so all your attack rolls are the same. However, you would get a +1 to damage from that, so add +1 to each one of those 1d8's except the one from vital strike.
Vital Strike would deal 2d8+1
Full-Attack #1 would deal 1d8+1 and 1d8+1 (from multishot)
Full-Attack #2 and #3 would each deal 1d8+1 if they hit.
Other static bonuses work the same way.
Lets say your bow is in fact a +1 bow, and you have favored enemy (orc) +2. Here's your attacks, assuming you attack an orc:
Attack action: d20+14 (2d8+3)
Full-Attack (Full-round action)
Attack #1: d20+14 (1d8+3 and 1d8+3)
Attack #2: d20+9 (1d8+3)
Attack #3 (Speed): d20+14 (1d8+3)
Those attack rolls are (BAB 6 + Dex 5 + Enhancement 1 + Favored 2) with attack #2 taking a -5 penalty since it's the iterative attack.
Then to be "legit" it's a +1 Speed Bow. You get the +1 to attack/damage and the speed property.
This is why GMs should never randomly roll for loot. It is game breaking.
Reminds me of a 3.5 home game I played in when the GM rolled off the old school 2nd Ed Loot tables. Rolled 2 %100 and then a %98. With out thinking about it further he handed the Half-orc with a 7 Int a Intelligent +3 Composite longbow of such and such with an Ego of something like 15. One failed save later, the bow won out, and that was the end of that home game.