\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{\frac {7}{2}}}+\frac {2 \left (A b -7 B a \right ) \sqrt {b \,x^{2}+a}}{21 a \,x^{\frac {3}{2}}}-\frac {2 b^{\frac {3}{4}} \left (A b -7 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {5}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (7 \, B a - A b\right )} \sqrt {b} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left ({\left (7 \, B a + 2 \, A b\right )} x^{2} + 3 \, A a\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )}}{21 \, a x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}}, x\right ) \]