$$int{(x-(\sqrt{x}-5)^2)^2} dx$$
I find the answer using expansion. How to solve it using substitution?
That is because of a small typo that HallsofIvy made. His method is spot on.
$\displaystyle \begin{align*}\int \big( x- (\sqrt{x}-5)^2 \big)^2 dx & = 2\int \big(u^2-(u-5)^2 \big)^2(udu) \\ & = 2\int (10u-25)^2udu \\ & = 50\int (4u^3-20u^2+25u)du \\ & = 50\left(u^4 - \dfrac{20}{3}u^3+\dfrac{25}{2}u^2\right) + C \\ & = 50x^2-\dfrac{1000}{3}x^{3/2} + 625x + C\end{align*}$
Halls just missed an exponent of 2.