\int{\frac{x^2+\sqrt{x+1}}{\sqrt{x+1}}}dx=\int{\frac{x^2dx}{\sqrt{x+1}}}+\int{dx}\\t=\sqrt{x+1}\; t^2=x+1\; x=t^2-1\; dx=2tdt\; x^2=(t^2-1)^2\\2\int{\frac{(t^2-1)^2tdt}{t}}+x=\cdots