\int\frac{dx}{\sqrt[4]{2x^2+x^4}}=\int\frac{dx}{x\sqrt[4]{1+\frac 2{x^2}}}=\int\frac{2x\,dx}{2x^2\sqrt[4]{1+\frac{2}{x^2}}}=\int\frac{dt}{2t\sqrt[4]{1+\frac 2t}}