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