Your problem has lots of powers of and multiples of (2+3t)^(1/3) in numerators and denominators and products and sums.

If you think very carefully, would it be possible for you to transform all your expressions into z==(2+3t)^(1/3) for powers and multiples of z and do that in a way that greatly simplifies your problem instead of making it more complicated? You are going to have to handle t very carefully to do this.

If you can correctly do that then is there any chance that Mathematica might be able to find a solution for you?