[Reincarnate]

1.40241865


Steps:

Rewrite/switch order to
(Integral from 0 to 2) (Integral from 0 to sqrt(y)) x*(y^2 + 2)^(1/4) dx dy

Solve:
(Integral from 0 to sqrt(y)) x*(y^2 + 2)^(1/4) dx
= (1/2)*y*(y^2 + 2)^(1/4)

Solve:
(Integral from 0 to 2) (1/2)*y*(y^2 + 2)^(1/4) dy
= (2/5) * 2^(1/4) * (3*3^(1/4)-1)

It helps sometimes to sketch the bounds out so you can see if something better can be used. The reason why keeping it in its original form is hard is because the inner part is a nonelementary integral, so that's the first major hint that you should probably try to rework the bounds/switch the order.