\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} \left (e x \right )^{\frac {9}{2}}}{c \,d^{2} e \sqrt {d \,x^{2}+c}}-\frac {\left (77 a^{2} d^{2}-198 a b c d +117 b^{2} c^{2}\right ) e \left (e x \right )^{\frac {5}{2}} \sqrt {d \,x^{2}+c}}{77 c \,d^{3}}+\frac {2 b^{2} \left (e x \right )^{\frac {9}{2}} \sqrt {d \,x^{2}+c}}{11 d^{2} e}+\frac {5 \left (77 a^{2} d^{2}-198 a b c d +117 b^{2} c^{2}\right ) e^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{231 d^{4}}-\frac {5 c^{\frac {3}{4}} \left (77 a^{2} d^{2}-198 a b c d +117 b^{2} c^{2}\right ) e^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{462 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {17}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (117 \, b^{2} c^{4} - 198 \, a b c^{3} d + 77 \, a^{2} c^{2} d^{2} + {\left (117 \, b^{2} c^{3} d - 198 \, a b c^{2} d^{2} + 77 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (42 \, b^{2} d^{4} x^{6} + 585 \, b^{2} c^{3} d - 990 \, a b c^{2} d^{2} + 385 \, a^{2} c d^{3} - 6 \, {\left (13 \, b^{2} c d^{3} - 22 \, a b d^{4}\right )} x^{4} + 2 \, {\left (117 \, b^{2} c^{2} d^{2} - 198 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {7}{2}}}{231 \, {\left (d^{6} x^{2} + c d^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]