The keys are the 13 x _ / _ and the _ x _ / _.
Those are the only terms that involve division, and since we need to come out with an integer in the end, we must either have those two terms each individually be integers, or they must both be fractions that add to an integer.
I focused on that second possibility, because that's the kind of sneaky thing people who make puzzles to do.
There are a couple ways to do this. I started with 13 x _ / 3 for the first and _ x 6 / 9 for the second.
Then I simply started going through the possibilities for 13 x _, and for each picking the smallest value using the remaining digits for _ x 6 / 9 that made the sum of these two terms an integer.
There are only 6 cases here. For each of those cases, you finally end up with something of the form _ + _ - _ + 12 x _ = I, where I is an integer. There will only be one or two possible values from the digits remaining that can go in the 12 x _ part and get you close enough to I that it is even worth continuing with that case. It's then trivial to see if the remaining 3 digits can be fit into _ + _ - _ to give what you need (and sometimes you can even skip that by doing a parity check).
Annoyingly, it is the last of the 6 cases that works, giving
1 + 13 x 8 / 3 + 7 + 12 x 4 - 5 - 11 + 2 x 6 / 9 - 10 = 66
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