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