I'm just looking at what you have there. Is sda5 your swap partition?
If you had somewhere you could copy the contents of sda6 (BACKUP) to you could possibly make it smaller and then just shuffle sda5 along, making your logical partitions smaller. But you very well could (will?) loose data from sda6. It might just be easier to backup up /var remove sda5 & 6 use gparted to make root(sda3) bigger then just put /var back in a smaller logical partition - being sure to have it still located at /sda6 you don't want things looking for it there if it ends up in another partition.