\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x}}{\sqrt {-b}}, \sqrt {-\frac {c}{d}}\right )}{\sqrt {-b}\, \sqrt {d}} \]
command
integrate((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, \sqrt {-b c d} c d {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right ) + \sqrt {-b c d} {\left (c + 2 \, d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right )}}{3 \, b c d^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}}{b d x^{2} + b x}, x\right ) \]