\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{11 c \,x^{\frac {11}{2}}}-\frac {2 a \left (-5 a d +22 b c \right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{77 c^{2} x^{\frac {7}{2}}}-\frac {2 \left (5 a^{2} d^{2}-22 a b c d +77 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{231 c^{2} x^{\frac {3}{2}}}+\frac {2 d^{\frac {3}{4}} \left (5 a^{2} d^{2}-22 a b c d +77 b^{2} c^{2}\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {x}}{c^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {x}}{c^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {x}}{c^{\frac {1}{4}}}\right )\right ) c^{\frac {9}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (77 \, b^{2} c^{2} - 22 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d} x^{6} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left ({\left (77 \, b^{2} c^{2} + 44 \, a b c d - 10 \, a^{2} d^{2}\right )} x^{4} + 21 \, a^{2} c^{2} + 6 \, {\left (11 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )}}{231 \, c^{2} x^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c}}{x^{\frac {13}{2}}}, x\right ) \]