The word between the number and the bonus is important. For each unique type, take the highest. If there's no word, add them. Be careful of the wording of bonuses, many of them changed when essentials came out to make them more compatible with weapons expertise.
If you have a +2 racial bonus to arcana, a +3 item bonus to Arcana, a +4 item bonus to arcana, then you have a total of +2+max(+3,+4)=+6 bonus.
Example 1: Hellfire blood has no word between the number and the bonus, so it adds to everything. Using a fire keyword power, in heroic, you get a +2 to attack: +1 feat, +1 untyped.
Hellfire blood:
Hellfire Blood Heroic Tier Prerequisite: Tiefling Benefit: You gain a
+1 bonus to attack rolls and damage rolls when you use a power that has the fire or the fear keyword.
The confusion over this feat stems from the Feb 2010 errata:
Update (2/8/2010)
On the Benefit line, delete “feat.” This change ensures that this feat continues to provide a benefit despite changes to the Weapon Expertise and Implement Expertise feats.
Example 2: Total +4 to arcana checks.
This actually does work together. Note the text:
You gain a +1 feat bonus to attack rolls you make with any weapon with which you have proficiency and with a wand or another item designated as a bard implement.
The first part gives you a bonus to attack rolls you make with any weapon with which you have proficiency. The second gives you bonuses with wands or other bard implements.
Note that it doesn't say "weapon attack rolls." Instead, it says "attack rolls you make with any weapon." They're not the same.
Now, the rules for Weapons as Implements and vice versa:
Using a Weapon as an Implement: If an adventurer is able to use a weapon as an implement, the weapon works like a normal implement, but the adventurer uses neither the weapon’s proficiency bonus nor its nonmagical weapon properties with his or her implement powers....
When an adventurer uses a magic version of the weapon as an implement, he or she can use the magic weapon’s enhancement bonus, critical hit effects, properties, and powers. However, some magic weapons have properties and powers that work only with weapon powers (Rules Compendium 275).
You're making an attack with a weapon with which you have proficiency. You happen to be using it as an implement, but that doesn't matter- the text says you get the bonus with attack rolls made with any weapon, and you're making an attack roll with a weapon.
This issue comes up in a variety of ways in the game. For example, if you're using a Frost Shortbow, you could attach a Siberys Shard of Merciless Cold to it to do extra damage, which would work even when making an implement attack. If you're using a Weapon of Speed Shortbow, you make make a Ranged Basic Attack with it once per encounter as a minor action...even if the RBA is using the bow as an implement, not as a weapon.
Best Answer
Battle Song Expertise should work with weaplements.
With the errata to the feat (see Patrick vD's answer), Battle Song Expertise now applies to "any weapon with which you are proficient". If you take Arcane Implement Proficiency for a kind of weapon, you will be proficient with it. It will also still be a weapon (even though you're using it as an implement). The feat does not limit its bonus to weapon powers, therefore you do gain the feat's bonus when using a weapon-as-implement.