\[ \int \frac {1}{(c x)^{3/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{3 a c \sqrt {c x}\, \sqrt {-2 a \,x^{2}+3 a}}+\frac {2^{\frac {1}{4}} \EllipticE \left (\frac {\sqrt {3-x \sqrt {6}}\, \sqrt {6}}{6}, \sqrt {2}\right ) \sqrt {c x}\, \sqrt {-2 x^{2}+3}\, 3^{\frac {1}{4}}}{3 a \,c^{2} \sqrt {x}\, \sqrt {-2 a \,x^{2}+3 a}}-\frac {\sqrt {-2 a \,x^{2}+3 a}}{3 a^{2} c \sqrt {c x}} \]
command
integrate(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (2 \, x^{3} - 3 \, x\right )} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} {\left (x^{2} - 1\right )}}{3 \, {\left (2 \, a^{2} c^{2} x^{3} - 3 \, a^{2} c^{2} x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{4 \, a^{2} c^{2} x^{6} - 12 \, a^{2} c^{2} x^{4} + 9 \, a^{2} c^{2} x^{2}}, x\right ) \]