Answer
The answer is in an implicit form :
$-t^{-1} + t^3/3 + C $
Work Step by Step
$du/dt = (1 + t^4)/(ut^2 + u^4t^2) $
This can be rewritten as:
$du/dt = (1 + t^4)/((t^2)(u + u^4)) $
Now it will be easier to separate u on one side and t on the other side:
$(u + u^4) du = (1 + t^4)/(t^2) dt $
This can be simplified to :
$(u + u^4) du = (t^{-2} + t^2) dt $
Now let's integrate both sides:
$\int(u + u^4) du = \int(t^{-2} + t^2) dt $
$u^2/2 + u^5/5 = -t^{-1} + t^3/3 + C $
