\[ \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}} \, dx \]
Optimal antiderivative \[ -\frac {9 \,3^{\frac {1}{4}} c^{2} \EllipticE \left (\frac {\sqrt {3-x \sqrt {6}}\, \sqrt {6}}{6}, \sqrt {2}\right ) \sqrt {c x}\, \sqrt {-2 x^{2}+3}\, 2^{\frac {1}{4}}}{10 \sqrt {x}\, \sqrt {-2 a \,x^{2}+3 a}}-\frac {c \left (c x \right )^{\frac {3}{2}} \sqrt {-2 a \,x^{2}+3 a}}{5 a} \]
command
integrate((c*x)^(5/2)/(-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
\[ -\frac {2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x - 9 \, \sqrt {2} \sqrt {-a c} c^{2} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{10 \, a} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x^{2}}{2 \, a x^{2} - 3 \, a}, x\right ) \]