Path of the Arcane Self

This class offers a bit of magic to the barbarian without needing to rely to much on the randomness of the Path of Wild Magic, the spellcasting of this subclass is very limited, even compared to other subclasses that give magic to martials, such as eldritch knight and arcane trickster. Because your are using the fortitude of your body to muster up magic, constitution is your spellcasting modifier.

Magic Strength

While raging, all of your attacks (including unarmed strikes) count as magical and bypass nonmagical resistances to slashing/piecing/bludgeoning, and you can add 1d4 of any magical damage type (fire, cold, lightning, radiant, necrotic, acid, poison, psychic, thunder, force etc) of your choice this damage increases to 1d6 at lvl 6, 1d8 at lvl 10, and 1d10 at lvl 14. additionally after you use this ability you can reduce incoming damage of the same type by to the number you rolled for the rest of the turn.

Enraged Spellcasting

While raging, you can cast spells with Constitution as your Spellcasting Ability. see chart for details of spell slots and known spells, you can change your known spells every dawn and dusk.

Spell List:

Cantrips: Control Flames, Create Bonfire, Fire Bolt, Frostbite, Gust, Light, Lightning Lure, Message, Poison Spray, Primal Savagery, Produce Flame, Resistance, Shape Water, Sword Burst, Thaumaturgy, Thorn Whip, Thunderclap, True Strike

1st Level: Absorb Elements, Armor of Agathy's, Bane, Catapult, Cause Fear, Chaos Bolt, Command, Compelled Duel, Earth Tremor, Ensnaring Strike, Entangle, Expeditious Retreat, Feather Fall, Grease, Jump, Longstrider, Magic Missile, Shield, Snare, Speak with Animals, Zephyr Strike

2nd Level: Animal Friendship, Beast sense, Blindness/Deafness,  Blur, Enhance Ability, Heat Metal, Hold Person, Knock, Magic Weapon, Spider Climb, Web

3rd Level: Call Lightning, Crusader's Mantle, Elemental Weapon, Haste, Slow, Stinking Cloud, Water Breathing

Flexible Magic

While raging, your concentration cannot be broken, additionally you can replace one of your attacks from Extra Attack with a spell (you must use be taking the attack action, you cannot cast 2 spells)

Arcane Overflow

The raw power of your magic is so strong that all your spells gain 1 free level of upcasting without costing a higher level spell slot. Additionally, any cantrips that normally cost an action now only cost a bonus action. Also due to your attunement to magic you have a +2 bonus to AC against magical attacks.