\[ \int \frac {(c x)^{3/2}}{\left (a+b x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {c \sqrt {c x}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {c \sqrt {c x}}{6 a b \sqrt {b \,x^{2}+a}}+\frac {c^{\frac {3}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{12 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) a^{\frac {5}{4}} b^{\frac {5}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((c*x)^(3/2)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )} \sqrt {b c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (b^{2} c x^{2} - a b c\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{6 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {c x} c x}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \]