Problem with indefinite integration
$$J:=\int \frac{2\sqrt[5]{2x-3}-1}{\left(2x-3\right)\sqrt[5]{2x-3}+\sqrt[5]{2x-3}}dx=\int\frac{dx}{x-1}-\frac12\int\frac{dx}{(x-1)\sqrt[5]{2x-3}}=$$
Substitute in the second integral as you did:
$$t^5=2x-3\implies dx=\frac{5t^4}2dt\implies J=\log|x-1|-\frac54\int\frac{t^4\,dt}{\frac{t^5+1}2t}=$$
$$=\log|x-1|-\frac52\int\frac{t^3}{t^5+1}dt$$
Now do partial fractions:
$$t^5+1=(t+1)(t^4-t^3+t^2-t+1)$$
To decompose that you better know the roots of unity of degree $\;5\;$