\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx \]
Optimal antiderivative \[ -\frac {3 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right ) \sqrt {2}}{4}+\frac {\left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x +\sqrt {x}\, \sqrt {1+x}}}{2} \]
command
integrate((-x+x^(1/2)*(1+x)^(1/2))^(1/2)/(1+x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{2} \, \sqrt {\sqrt {x + 1} \sqrt {x} - x} {\left (3 \, \sqrt {x + 1} + \sqrt {x}\right )} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {2 \, {\left (13230146471646497941753920 \, \sqrt {2} {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {x} + 472818412040 \, \sqrt {2} {\left (111925814517792 \, x^{2} + 83944360888344 \, x + 621904154881\right )} + 3497681703681 \, {\left (\sqrt {2} {\left (50987674019848 \, x - 621904154881\right )} \sqrt {x + 1} + \sqrt {2} {\left (60938140497944 \, x - 24871932855043\right )} \sqrt {x}\right )} \sqrt {\sqrt {x + 1} \sqrt {x} - x}\right )}}{139214258174109988596285504 \, x^{2} - 678551053586366160645570576 \, x + 386764777858250836124161}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]