\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \sqrt {d \,x^{2}+c}}{7 c e \left (e x \right )^{\frac {7}{2}}}-\frac {2 a \left (-5 a d +14 b c \right ) \sqrt {d \,x^{2}+c}}{21 c^{2} e^{3} \left (e x \right )^{\frac {3}{2}}}+\frac {\left (5 a^{2} d^{2}-14 a b c d +21 b^{2} c^{2}\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {9}{4}} d^{\frac {1}{4}} e^{\frac {9}{2}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (21 \, b^{2} c^{2} - 14 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, a^{2} c d + {\left (14 \, a b c d - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {9}{2}\right )}}{21 \, c^{2} d x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d e^{5} x^{7} + c e^{5} x^{5}}, x\right ) \]