11.20 Problem number 608

\[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx \]

Optimal antiderivative \[ \frac {6^{\frac {3}{4}} a \,c^{\frac {3}{2}} \EllipticF \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {c x}}{3 \sqrt {c}}, i\right ) \sqrt {-2 x^{2}+3}}{7 \sqrt {a \left (-2 x^{2}+3\right )}}+\frac {2 \left (c x \right )^{\frac {5}{2}} \sqrt {-2 a \,x^{2}+3 a}}{7 c}-\frac {2 c \sqrt {c x}\, \sqrt {-2 a \,x^{2}+3 a}}{7} \]

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

\[ -\frac {3}{7} \, \sqrt {2} \sqrt {-a c} c {\rm weierstrassPInverse}\left (6, 0, x\right ) + \frac {2}{7} \, \sqrt {-2 \, a x^{2} + 3 \, a} {\left (c x^{2} - c\right )} \sqrt {c x} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c x, x\right ) \]