Karnaugh map with don't care: increasing the number of groups instead of simplifying AB
        00  01  11  10     
    00 | x | 1 | 0 | 1 |
CD  01 | 0 | 1 | x | 0 |
    11 | 1 | x | x | 0 |
    10 | x | 0 | 0 | x |


The answer to the above Karnaugh map is $F(ABCD) = B D + \bar B \bar D + \bar A \bar C \bar D + \bar A C D$
according to my book and the K-map solvers online.

But what I don't get is that I can further reduce the terms in this answer:
$F(ABCD) = \bar B \bar D + \bar A CD + \bar A B \bar C$
also covers all the min terms in lesser groups.

The answer in my book just has a bigger group covering additional don't cares. Since they are optional shouldn't my answer be correct?
Your answer is correct. It is also more reduced. The reason your book and the solver give you a bigger equation is because they use a greedy algorithm that attempts to match bigger groups (groups with less variable in them). This will have an optimal behavior if the map has no "don't cares" [reference needed].

To have optimal behavior with "don't care", you have to consider that an X can be either a 1 or a 0. That means that for each X, you have two versions of you map. So, if you have $N$ "don't cares", you will have to reduce the $2^N$ versions of the map and choose the smallest reduced operation from the results.

Extra

If you have lots of those maps to reduce, i suggest you use Logic Friday or a similar equation reducer.