Problem with simple integral I'm trying to solve this simple integral:
$$\frac12 \int \frac{x^2}{\sqrt{x + 1}} dx$$
Here's what I have done so far:


*

*$\displaystyle t = \sqrt{x + 1} \Leftrightarrow x = t^2 - 1 \Rightarrow dx = 2t dt$

*$\displaystyle \frac12 \int \frac{x^2}{\sqrt{x + 1}} dx = \int \frac{t (t^2 - 1)^2}t dt$

*$\displaystyle \int (t^2 - 1)^2 dt = \frac15 t^5 - \frac23 t^3 + t + C$

*$\displaystyle \frac15 t^5 - \frac23 t^3 + t + C = \frac15 \sqrt{(x + 1)^5} - \frac23 \sqrt{(x + 1)^3} + \sqrt{x + 1} + C$
WolframAlpha tells me steps 1 and 3 are right so the mistake must be somewhere in steps 2 and 4, but I really can't see it.

$$\begin{align}\frac{x^2}{\sqrt{x+1}} &= \frac{x^2-1}{\sqrt{x+1}} + \frac{1}{\sqrt{x+1}}\\&=(x-1)\sqrt{x+1} + \frac{1}{\sqrt{x+1}}\\&=(x+1)^{3/2} - 2\sqrt{x+1} + \frac{1}{\sqrt{x+1}}\end{align}$$
I think you can integrate each of  these terms.
