\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2}}{5 c e \left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {\left (77 a^{2} d^{2}-70 a b c d +5 b^{2} c^{2}\right ) \left (e x \right )^{\frac {3}{2}}}{15 c^{3} e^{5} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 a \left (-11 a d +10 b c \right )}{5 c^{2} e^{3} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {e x}}+\frac {\left (77 a^{2} d^{2}-70 a b c d +5 b^{2} c^{2}\right ) \left (e x \right )^{\frac {3}{2}}}{10 c^{4} e^{5} \sqrt {d \,x^{2}+c}}-\frac {\left (77 a^{2} d^{2}-70 a b c d +5 b^{2} c^{2}\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{10 c^{4} e^{4} \sqrt {d}\, \left (\sqrt {c}+x \sqrt {d}\right )}+\frac {\left (77 a^{2} d^{2}-70 a b c d +5 b^{2} c^{2}\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {15}{4}} d^{\frac {3}{4}} e^{\frac {7}{2}} \sqrt {d \,x^{2}+c}}-\frac {\left (77 a^{2} d^{2}-70 a b c d +5 b^{2} c^{2}\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{20 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {15}{4}} d^{\frac {3}{4}} e^{\frac {7}{2}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (3 \, {\left ({\left (5 \, b^{2} c^{2} d^{2} - 70 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{7} + 2 \, {\left (5 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 77 \, a^{2} c d^{3}\right )} x^{5} + {\left (5 \, b^{2} c^{4} - 70 \, a b c^{3} d + 77 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, b^{2} c^{2} d^{2} - 70 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{6} - 12 \, a^{2} c^{3} d + 5 \, {\left (5 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 77 \, a^{2} c d^{3}\right )} x^{4} - 12 \, {\left (10 \, a b c^{3} d - 11 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{30 \, {\left (c^{4} d^{3} x^{7} + 2 \, c^{5} d^{2} x^{5} + c^{6} d x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{3} e^{4} x^{10} + 3 \, c d^{2} e^{4} x^{8} + 3 \, c^{2} d e^{4} x^{6} + c^{3} e^{4} x^{4}}, x\right ) \]