\[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-5 A c +3 b B \right ) x^{\frac {3}{2}}}{3 b^{2} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {2 A}{3 b \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {\left (-5 A c +3 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} c^{\frac {1}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)*x^(1/2)/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{5} + {\left (3 \, B b^{2} - 5 \, A b c\right )} x^{3}\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - \sqrt {c x^{4} + b x^{2}} {\left (2 \, A b c - {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2}\right )} \sqrt {x}}{3 \, {\left (b^{2} c^{2} x^{5} + b^{3} c x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{8} + 2 \, b c x^{6} + b^{2} x^{4}}, x\right ) \]