\[ \int \frac {1}{(c x)^{5/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {5 \,2^{\frac {3}{4}} \EllipticF \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {c x}}{3 \sqrt {c}}, i\right ) \sqrt {-2 x^{2}+3}\, 3^{\frac {3}{4}}}{81 a \,c^{\frac {5}{2}} \sqrt {a \left (-2 x^{2}+3\right )}}+\frac {1}{3 a c \left (c x \right )^{\frac {3}{2}} \sqrt {-2 a \,x^{2}+3 a}}-\frac {5 \sqrt {-2 a \,x^{2}+3 a}}{27 a^{2} c \left (c x \right )^{\frac {3}{2}}} \]
command
integrate(1/(c*x)^(5/2)/(-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
\[ -\frac {5 \, \sqrt {2} {\left (2 \, x^{4} - 3 \, x^{2}\right )} \sqrt {-a c} {\rm weierstrassPInverse}\left (6, 0, x\right ) + 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} {\left (5 \, x^{2} - 3\right )}}{27 \, {\left (2 \, a^{2} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{4 \, a^{2} c^{3} x^{7} - 12 \, a^{2} c^{3} x^{5} + 9 \, a^{2} c^{3} x^{3}}, x\right ) \]