\[ \int \frac {1}{\sqrt {c x} \left (3 a-2 a x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {c x}}{3 \sqrt {c}}, i\right ) \sqrt {-2 x^{2}+3}\, 6^{\frac {3}{4}}}{18 a \sqrt {c}\, \sqrt {a \left (-2 x^{2}+3\right )}}+\frac {\sqrt {c x}}{3 a c \sqrt {-2 a \,x^{2}+3 a}} \]
command
integrate(1/(c*x)^(1/2)/(-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
\[ -\frac {\sqrt {2} \sqrt {-a c} {\left (2 \, x^{2} - 3\right )} {\rm weierstrassPInverse}\left (6, 0, x\right ) + 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{6 \, {\left (2 \, a^{2} c x^{2} - 3 \, a^{2} c\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{4 \, a^{2} c x^{5} - 12 \, a^{2} c x^{3} + 9 \, a^{2} c x}, x\right ) \]