\[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{5 b \,x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {3 \left (-7 A c +5 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right ) \sqrt {c}}{5 b^{3} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {\left (-7 A c +5 b B \right ) \sqrt {x}}{5 b^{2} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {3 \left (-7 A c +5 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{3} x^{\frac {3}{2}}}-\frac {3 c^{\frac {1}{4}} \left (-7 A c +5 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {3 c^{\frac {1}{4}} \left (-7 A c +5 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/(c*x^4+b*x^2)^(3/2)/x^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, {\left ({\left (5 \, B b c - 7 \, A c^{2}\right )} x^{6} + {\left (5 \, B b^{2} - 7 \, A b c\right )} x^{4}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, B b c - 7 \, A c^{2}\right )} x^{4} + 2 \, A b^{2} + 2 \, {\left (5 \, B b^{2} - 7 \, A b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{5 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{9} + 2 \, b c x^{7} + b^{2} x^{5}}, x\right ) \]