There are three ways you can save a route.
A button at the bottom
If you scroll down to the bottom of the results, you will find a ‘Save Route’ button.
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Tapping on the info box
If you tap on the info box at the top of the results, it will bring up a menu with an option to save the route.
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Dragging open a Save button
And from the Routes tab, if you drag from left to right on a recent route, it will reveal a Save button.

The delete button
Conversely, if you drag from right to left on a recent route, it will reveal a Delete button to remove that route from your recent routes.