$$\int_{}^{} \frac{x^5}{\sqrt[3]{1+x^3}}dx$$
$$\int_{}^{} \frac{x^5}{\sqrt[3]{1+x^3}}dx=\frac{1}{3}\int_{}^{} \frac{x^3}{\sqrt[3]{1+x^3}}dx^3=$$
$$=\frac{1}{3}\int_{}^{} \frac{x^3+1}{\sqrt[3]{1+x^3}}d(x^3+1)-\int_{1}^{3} \int_{}^{} \frac{1}{\sqrt[3]{1+x^3}}d(x^3+1)=$$
$$=\frac{1}{3}\int_{}^{} (1+x^3)^{2/3}d(x^3+1)-\frac{1}{3}\int_{}^{}(1+x^3)^{-1/3}d(x^3+1)=$$
$$=\frac{1}{5}(1+x^3)^{5/3}-\frac{1}{2}(+1x^3)^{2/3}+C=\sqrt[3]{(1+x^3)^2}[0,2x^3-0,3]+C$$
$$\sqrt[3]{(1+x^3)^2}[0,2x^3-0,3]+C$$