\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} \left (e x \right )^{\frac {9}{2}}}{3 c \,d^{2} e \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {\left (7 a^{2} d^{2}-42 a b c d +39 b^{2} c^{2}\right ) e \left (e x \right )^{\frac {5}{2}}}{14 c \,d^{3} \sqrt {d \,x^{2}+c}}+\frac {2 b^{2} \left (e x \right )^{\frac {9}{2}}}{7 d^{2} e \sqrt {d \,x^{2}+c}}-\frac {5 \left (7 a^{2} d^{2}-42 a b c d +39 b^{2} c^{2}\right ) e^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{42 c \,d^{4}}+\frac {5 \left (7 a^{2} d^{2}-42 a b c d +39 b^{2} c^{2}\right ) e^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{84 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {1}{4}} d^{\frac {17}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, {\left (39 \, b^{2} c^{4} - 42 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2} + {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (12 \, b^{2} d^{4} x^{6} - 195 \, b^{2} c^{3} d + 210 \, a b c^{2} d^{2} - 35 \, a^{2} c d^{3} - 4 \, {\left (13 \, b^{2} c d^{3} - 14 \, a b d^{4}\right )} x^{4} - 7 \, {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {7}{2}}}{42 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \]