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