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I would like to compute the integral $$ \int y\,dx, \qquad y=\sqrt{x+\sqrt{x+1}},\\ F(x,y)=y^4-2xy^2+x^2-x-1=0, $$ in closed form, where $F(x,y)$ is a polynomial in $\mathbb{C}[x,y]$.

I am trying to apply the Risch algorithm (from "Algorithms for Computer Algebra" by Geddes, Czapor and Labahn) by hand to integrate this in terms of elementary functions, but I don't understand it. I need to write the integrand $f(x,y)=y$ in the form $a(x,y)/d(x)$ with $d(x)$ square-free, so $a=y$ and $d=1$. Then the zeros of $d$ determine the poles of $f$ and the logarithmic components of the integral, so since $d$ has no zeros, either there is no logarithmic component or the integral is not elementary (I am not sure which).

Am I missing something obvious? The integral has a closed form: $$\int y\,dx = \frac{-3+8x+2\sqrt{1+x}}{12}\sqrt{x+\sqrt{x+1}} + \frac{5}{8}\log\left( 1+2\sqrt{x+1} + 2\sqrt{x+\sqrt{x+1}} \right). $$

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You should first find a basis for your function and then apply hermite reduction (p. 564 in your book)

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