First of all,
Since there are no stretches of 4 or 6 in any of the pieces, the 7 can only be 2+5 or 1+3+3. Also, we can see that no stretch of 3 fits the 3 squares north of the 7 without filling either the 2 or the 3 square.
Also, the only other choices for 3+3 around the 7 are west and south, and there is no way to fit a piece to complete the southward part without touching the westward part.
By contradiction, we can deduce that the 7 square has to be made up of 2+5.
There are only two pieces with a stretch of 5, so let's try all their positions. A northward three-piece would make it impossible to complete the 3 with any of the remaining pieces.
Same story with a northward seven-piece in either direction.
So the stretch of five needs to be westward. Note that we cannot fit the two-piece into any of the stretches of 2 required, so the stretch of 2 has to be filled in by a seven-piece. That leaves only the three-piece for the stretch of 5.
If we put the three-piece facing south, there is only one way to complete the 7, but this puts a stretch of 5 next to the 3-square.
Thus we conclude that the three-piece must face north. There is exactly one way to complete the 7 now, and we've completed the 2 in the process as well.
We know the 3 cannot be 2+1, because there is no way to complete the 3 if we use the stretch of two in either direction.
So the 3 has to be a stretch of 3 to the north. We have two pieces that fit that stretch of three. Let's try the two-piece first.
If we touch the five with a stretch of 3, we can't complete it with the remaining pieces which has no stretch of 2.
Putting the two-piece the other way, we would need a stretch of 5 to finish the 5-square.
So we have to use the five-piece for the 3, and there are two ways to do it. Let's try the wrong way first – this would require a stretch of 4 to complete the 5-square.
Only one position we haven't tried:
From there, the remaining piece only fits one way to complete the 5: