How to recover data from a partition with unknown filesystem

Question

I wanted to repartition my hard disk using KDE Partition Manager running from a USB. Unfortunately, something went wrong, as most of the partitions I wanted to delete are still present and the one I wanted to keep (but with larger size) has lost its filesystem information and, hence, it is unaccessible.

This is how it looks now sda5:

And that is fdisk -l output:

Disk /dev/sda: 596,2 GiB, 640135028736 bytes, 1250263728 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: dos
Disk identifier: 0x1549f232

Dispositiu Arrencada      Start      Final   Sectors   Size Id Tipus
/dev/sda1  *                 63  540619379 540619317 257,8G  7 HPFS/NTFS/exFAT
/dev/sda2            1224201195 1250258624  26057430  12,4G  7 HPFS/NTFS/exFAT
/dev/sda3             540619441 1224200012 683580572   326G  5 Estesa
/dev/sda5             540620863 1126558362 585937500 279,4G 83 Linux

If I want to change sda5 to its ext4 original filesystem, I get a warning stating that all changes will be lost.

Is there any way to recover original data or is it already lost for good?

Edit: as per @damadam request, this is the lsblk output:

NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT
sda      8:0    0 596,2G  0 disk 
├─sda1   8:1    0 257,8G  0 part 
├─sda2   8:2    0  12,4G  0 part 
├─sda3   8:3    0     1K  0 part 
└─sda5   8:5    0 279,4G  0 part 
sdb      8:16   0 232,9G  0 disk 
├─sdb1   8:17   0   450M  0 part 
├─sdb2   8:18   0    99M  0 part 
├─sdb3   8:19   0    16M  0 part 
├─sdb4   8:20   0 111,8G  0 part 
├─sdb5   8:21   0  27,5G  0 part /
└─sdb6   8:22   0  93,1G  0 part /home
sr0     11:0    1  1024M  0 rom  
loop0    7:0    0  86,9M  1 loop /snap/core/4917
loop1    7:1    0  86,6M  1 loop /snap/core/4650
loop2    7:2    0  86,9M  1 loop /snap/core/4830

Answer 1

For further reference: Following @MaNyYaCk's advice, I have managed to recover the partition using Testdisk

1. Download from https://www.cgsecurity.org/wiki/TestDisk_Download and uncompress
2. Locate the folder download on #1 and run command sudo ./testdisk_static
3. Click on create
4. Navigate through the hard drive(s) and select the desired one and then the command Proceed.
5. Select the type of partitions, which in my case was Intel (usually makes a reasonable guess and selects it accordingly)
6. Select the partitions you want to work with and then select Analyse command.
7. Select Quick Search command. After a while (it can take some time) it will find the partitions on that drive and you will see new options available.
8. Click on the new write option that appeared after #7 (if it is not available you may want to make a Deep search instead of a quick one). You will be asked to confirm that you want to write partition table.
9. If everything goes as expected your partitions will be recovered and available again (it may require to reboot)