\[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 \,2^{\frac {1}{4}} \EllipticE \left (\frac {\sqrt {3-x \sqrt {6}}\, \sqrt {6}}{6}, \sqrt {2}\right ) \sqrt {c x}\, \sqrt {-2 x^{2}+3}\, 3^{\frac {1}{4}}}{3 c^{2} \sqrt {x}\, \sqrt {-2 a \,x^{2}+3 a}}-\frac {2 \sqrt {-2 a \,x^{2}+3 a}}{3 a c \sqrt {c x}} \]
command
integrate(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\sqrt {2} \sqrt {-a c} x {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, a c^{2} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{2 \, a c^{2} x^{4} - 3 \, a c^{2} x^{2}}, x\right ) \]