15.10 Problem number 1013

$$\int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}}dx$$

Optimal antiderivative $$\mathrm{arccosh}\left(\sqrt{x}\right)+\sqrt{x}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}$$

command

integrate(x**(1/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$\frac{G_{6,6}^{6,2}\left(\begin{array}{cc}-\frac{3}{4},-\frac{1}{4}& -\frac{1}{2},-\frac{1}{2},0,1\\ -1,-\frac{3}{4},-\frac{1}{2},-\frac{1}{4},0,0& \end{array}\right|\frac{1}{x})}{2{\pi }^{\frac{3}{2}}}-\frac{i{G}_{6,6}^{2,6}\left(\begin{array}{cc}-\frac{3}{2},-\frac{5}{4},-1,-\frac{3}{4},-\frac{1}{2},1& \\ -\frac{5}{4},-\frac{3}{4}& -\frac{3}{2},-1,-1,0\end{array}\right|\frac{e^{2i\pi }}{x})}{2{\pi }^{\frac{3}{2}}}$$