\[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 \,2^{\frac {3}{4}} a \EllipticF \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {c x}}{3 \sqrt {c}}, i\right ) \sqrt {-2 x^{2}+3}\, 3^{\frac {3}{4}}}{9 c^{\frac {5}{2}} \sqrt {a \left (-2 x^{2}+3\right )}}-\frac {2 \sqrt {-2 a \,x^{2}+3 a}}{3 c \left (c x \right )^{\frac {3}{2}}} \]
command
integrate((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {2} \sqrt {-a c} x^{2} {\rm weierstrassPInverse}\left (6, 0, x\right ) - \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, c^{3} x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{c^{3} x^{3}}, x\right ) \]