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