\int \left(x^2+\frac{\sqrt{x+1}}{\sqrt[3]{x+1}}\right)\text{dx}=\frac{x^3}{3}+\int (x+1)^{\frac{1}{2}-\frac{1}{3}}\text{dx}=\frac{x^3}{3}+\int (x+1)^{\frac{1}{6}}\text{dx}=\frac{x^3}{3}+\frac{6}{7} (1+x)^{7/6}+c