\[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {c \left (c x \right )^{\frac {3}{2}}}{2 a \sqrt {-2 a \,x^{2}+3 a}}+\frac {3 \,3^{\frac {1}{4}} c^{2} \EllipticE \left (\frac {\sqrt {3-x \sqrt {6}}\, \sqrt {6}}{6}, \sqrt {2}\right ) \sqrt {c x}\, \sqrt {-2 x^{2}+3}\, 2^{\frac {1}{4}}}{4 a \sqrt {x}\, \sqrt {-2 a \,x^{2}+3 a}} \]
command
integrate((c*x)^(5/2)/(-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
\[ -\frac {2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x + 3 \, \sqrt {2} {\left (2 \, c^{2} x^{2} - 3 \, c^{2}\right )} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{4 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x^{2}}{4 \, a^{2} x^{4} - 12 \, a^{2} x^{2} + 9 \, a^{2}}, x\right ) \]