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