\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \EllipticE \left (\frac {\sqrt {c}\, \sqrt {b x}}{\sqrt {-b}}, i\right )}{\sqrt {-b}\, \sqrt {c}} \]
command
integrate((-c*x+1)^(1/2)/(b*x)^(1/2)/(c*x+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\sqrt {-b c^{2}} c {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) + \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right )}}{b c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}}{b c x^{2} + b x}, x\right ) \]