\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{c \sqrt {c x}}+\frac {12 b \left (c x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}{5 c^{3}}+\frac {24 a \sqrt {b}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}{5 c^{2} \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {24 a^{\frac {5}{4}} b^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) c^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}+\frac {12 a^{\frac {5}{4}} b^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) c^{\frac {3}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((b*x^2+a)^(3/2)/(c*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, \sqrt {b c} a x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - \sqrt {b x^{2} + a} {\left (b x^{2} - 5 \, a\right )} \sqrt {c x}\right )}}{5 \, c^{2} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}}{c^{2} x^{2}}, x\right ) \]