Aloha :)
$$I=\int\frac{1}{\sqrt[4]{x^5}-\sqrt[4]{x^3}}\,dx=\int\frac{1}{\left(\sqrt[4]{x}\right)^5-\left(\sqrt[4]{x}\right)^3}\,dx$$Substituiere wie folgt:$$u\coloneqq\sqrt[4]{x}=x^{\frac14}\quad\implies\quad\frac{du}{dx}=\frac14x^{-\frac34}=\frac{1}{4\left(\sqrt[4]{x}\right)^3}=\frac{1}{4u^3}\implies dx=4u^3\,du$$Damit vereinfacht sich das Integral zu:$$I=\int\frac{1}{u^5-u^3}\,4u^3\,du=\int\frac{4}{u^2-1}\,du=2\int\frac{2}{(u-1)(u+1)}\,du$$$$\phantom{I}=2\int\left(\frac{1}{u-1}-\frac{1}{u+1}\right)du=2\left(\ln|u-1|-\ln|u+1|\right)+\text{const}$$$$\phantom{I}=2\ln\left|\frac{u-1}{u+1}\right|+\text{const}=2\ln\left|1-\frac{2}{u+1}\right|+\text{const}=2\ln\left|1-\frac{2}{\sqrt[4]{x}+1}\right|+\text{const}$$
