The solution to the 3x3x5 cube is interesting. Initially I approached it by doing the inner layers as a 3x3x3, the outer layers as a domino and then considering the entire cube as a 3x3x3 again to correct issues. This was really slow! Instead Graharg and I came up with a fairly nice reduction method which makes it only a bit slower than a regular 3x3x3.

Step 1

Gather together enough pieces so that one extended layer can turn. You don't need to reshape the whole thing back to a fully turnable state, only enough to turn one layer. Also, the colours or where the layer is really don't matter. This is usually doable in less than ten turns I'd say, often a lot less, but it does depend on the scramble.

I consider the turnable outer layer to be on the U face.

Step 2

The next step is to pair up at least 4 edges. This is very easy. Rotate the top outer layer until you see an edge that matches with another. Exchange it with an edge somewhere else in the puzzle that is not solved. Repeat until you have paired up 4 edges (meaning that there are none left to exchange).

Step 3

Now you will have a number of unsolved edge pairs and nowhere to exchange them out. I'll split this up in to the number of unsolved edge pairs because each has a different solution.

0. If you have zero unsolved edges then this step is already complete. Well done!
1. You can't have just one unsolved edge, that doesn't make sense.
2. If you have two unsolved edges, position them so that they are opposite each other on the top layer, and then put edges with matching colours in the other two edge positions. Now when you do a 180 twist all the edges will solve.
3. Sometimes you have three unmatched, although this seems to be a bit rarer. Two of them will require a 90 degree twist to solve, one will require a 180 degree twist. That 180 degree edge is the one you want to focus on. Substitute an edge for the solved edge which matches the colour of the top part of the 180 edge. Now a 90 degree twist will solve all of them.
4. If these aren't solve by some upper layer twist, there are two possible reasons. Both need a minor adjustment to be solvable with an upper layer twist.
4a. If the edges are 4-cyclic, it's because there is a Z-shaped permutation of the edges. Find an edge that needs to be paired with one a quarter twist away and then swap them.
4b. If the edges have two 2-cycles, make sure the cyclic pairs are on opposite sides of the cube.

Step 4

Now all the edges are solved, we want to set up a special symmetry on the edges. Make sure the opposing edges on the top layer have matching colours. E.g. you might have red-red/yellow oppposite red-red/white and blue-blue/white opposite blue-blue/yellow. The actual colours are irrelevent as long as they match opposite.

Step 5

Solve the corners in the same way you solve the edges; make matching corners until you are reduced to cycles. The only issue is that you must be left with exactly four incorrect corners. If you have less than four, you'll need to increase it until you have precisely four by unmatching some corners.

Once you have four corners you need to check how the cycles are laid out. This is almost exactly the same as step 3-4 above. You should be able to solve all four corners at once, to give you all eight corners matching.

Step 6

Sometimes this leaves the edges in an unmatched state. Because of the symmetry set up in step 4, there is only one possible case, and the algorithm to solve it is M2 U M' D2 M U' M2 Q' M2 D2 M2 Q where Q is a clockwise turn of the top outer layer, M is a downwards turn of the vertical central layer facing you, and the other moves are analogous to the 3x3x3.

Step 7

Solve as you would a 3x3x3.

The end

This method seems like a lot of steps but most of them are very quick. It's also very easy to remember with only one algorithm to remember. Still, it may be faster to solve like a 3x3x3 then a domino if you are a very fast solver.