Since is the substitution being made, you can take a fourth root on both sides to get . The is important here, and is why the integral needs to be split into two regions.
In the region , we should get because x is always negative in this region. Using the same logic, for , we should use . Now that you have an expression for x in both regions you can make the substitution.
By the way, make sure you use parentheses in your integral. It should be like this: