I have:
$$\int\frac{\sqrt[4]{3x+5}}{x}\,\mathrm{d}x = \\ = \begin{array}{|cc|}\sqrt[4]{3x+5} = t \\ t^4 = 3x+5 \\ 4t^3\mathrm{d}t = 3\mathrm{d}x \\ \mathrm{d}x = \frac{4}{3}t^3\mathrm{d}t \end{array} = 4\int\frac{t^4\mathrm{d}t}{t^4-5}$$
From there I tried to add and substract $5$ in the nunerator but do not know how to deal with $$5\int\frac{\mathrm{d}t}{t^4-5}$$
Now, write $$\frac{1}{t^4-5}=\frac{1}{(t^2-\sqrt5)(t^2+\sqrt5)}=\frac{1}{2\sqrt5(t^2-\sqrt5)}-\frac{1}{2\sqrt5(t^2+\sqrt5)}=$$ $$=\frac{1}{2\sqrt5(t-\sqrt[4]5)(t+\sqrt[4]5)}-\frac{1}{2\sqrt5(t^2+\sqrt5)}=$$ $$=\frac{1}{4\sqrt[4]{125}(t-\sqrt[4]5)}-\frac{1}{4\sqrt[4]{125}(t+\sqrt[4]5)}-\frac{1}{2\sqrt5(t^2+\sqrt5)}.$$