\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{\frac {9}{2}}}+\frac {2 \left (A b -3 B a \right ) \sqrt {b \,x^{2}+a}}{15 a \,x^{\frac {5}{2}}}+\frac {4 b \left (A b -3 B a \right ) \sqrt {b \,x^{2}+a}}{15 a^{2} \sqrt {x}}-\frac {4 b^{\frac {3}{2}} \left (A b -3 B a \right ) \sqrt {x}\, \sqrt {b \,x^{2}+a}}{15 a^{2} \left (\sqrt {a}+x \sqrt {b}\right )}+\frac {4 b^{\frac {5}{4}} \left (A b -3 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \sqrt {b \,x^{2}+a}}-\frac {2 b^{\frac {5}{4}} \left (A b -3 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (6 \, {\left (3 \, B a b - A b^{2}\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (6 \, {\left (3 \, B a b - A b^{2}\right )} x^{4} + 5 \, A a^{2} + {\left (9 \, B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )}}{45 \, a^{2} x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {11}{2}}}, x\right ) \]