\[ \int \frac {1}{(c x)^{5/2} \sqrt {3 a-2 a x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 \,2^{\frac {3}{4}} \EllipticF \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {c x}}{3 \sqrt {c}}, i\right ) \sqrt {-2 x^{2}+3}\, 3^{\frac {3}{4}}}{27 c^{\frac {5}{2}} \sqrt {a \left (-2 x^{2}+3\right )}}-\frac {2 \sqrt {-2 a \,x^{2}+3 a}}{9 a c \left (c x \right )^{\frac {3}{2}}} \]
command
integrate(1/(c*x)^(5/2)/(-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
\[ -\frac {2 \, {\left (\sqrt {2} \sqrt {-a c} x^{2} {\rm weierstrassPInverse}\left (6, 0, x\right ) + \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{9 \, a c^{3} x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{2 \, a c^{3} x^{5} - 3 \, a c^{3} x^{3}}, x\right ) \]