During Christ's time and before and after, they would read from the Septuagint "version". Most likely the text quoted in the OP is from the Masoretic "version". As well, not all translations are word for word. With these things in mind, here are the two texts from Luke and the Septuagint of Isaiah.
Isaiah 61:1-2 The Spirit of the Lord is upon me,
because he has anointed me; he has sent me
to preach glad tidings to the poor,
to heal the broken in heart,
to proclaim liberty to the captives,
and recovery of sight to the blind;
61:2 to declare the acceptable year of the Lord, [the remainder not read]
Luke 4:18-19 The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, To preach the acceptable year of the Lord.
Although "columns" don't align, here is a comparison of the two versions.
Luke 4:18 The Spirit of the Lord is upon me, Isaiah61:1 The Spirit of the Lord is upon me,
because he hath anointed me because he has anointed me; he has sent me
- to preach the gospel to the poor;
he hath sent me to preach glad tidings to the poor,
- to heal the brokenhearted, to heal the broken in heart,
- to preach deliverance to the captives, to proclaim liberty to the captives,
- and recovering of sight to the blind, and recovery of sight to the blind;
LK 4:18 to set at liberty them that are bruised,
4:19 To preach the acceptable year of the Lord. 61:2 to declare the acceptable year of the Lord,
Except for one clause in Luke 4:18 (to set at liberty them that are bruised), the remainder clauses line up more or less exactly.
So, what about the one clause? In my opinion, it relates to why Christ did not finish all of Isaiah.
Septuagint and the day of recompence; to comfort all that mourn;
Masoretic and the day of vengeance of our God; to comfort all that mourn;
To set at liberty the bruised means "The proclaiming of perfect liberty to the bound, and the year of acceptance with Jehovah, is a manifest allusion to the proclaiming of the year of the jubilee by sound of trumpet; and our Saviour, by applying this text to himself, plainly declares the typical design of that institution. Isa 42:7; Isa 49:9,24,25; Psa 102:20; Jer 34:8; Zec 9:11,12; Jhn 8:32-36; Act 26:18; Rom 6:16-22; Rom 7:23-25; 2Ti 2:25,26" -source-
In other words, Christ was preaching in Luke something not fully explained in the Old. But the rejection was; that is, the day of vengeance. That, however, was not related to Christ's first appearance, but rather His second coming. To set at liberty the bruised was tied to His first coming. Hence it was included in Luke, but not Isaiah and the remainder of Isaiah was not quoted or included in Luke.