\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (57 a^{2} d^{2}+b c \left (-38 a d +9 b c \right )\right ) \left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{627 d^{2} e}-\frac {2 b \left (-38 a d +9 b c \right ) \left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{285 d^{2} e}+\frac {2 b^{2} \left (e x \right )^{\frac {9}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{19 d \,e^{3}}+\frac {4 c \left (57 a^{2} d^{2}+b c \left (-38 a d +9 b c \right )\right ) \left (e x \right )^{\frac {5}{2}} \sqrt {d \,x^{2}+c}}{1463 d^{2} e}+\frac {8 c^{2} \left (57 a^{2} d^{2}+b c \left (-38 a d +9 b c \right )\right ) e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{4389 d^{3}}-\frac {4 c^{\frac {11}{4}} \left (57 a^{2} d^{2}+b c \left (-38 a d +9 b c \right )\right ) e^{\frac {3}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{4389 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {13}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((e*x)^(3/2)*(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
\[ -\frac {2 \, {\left (20 \, {\left (9 \, b^{2} c^{5} - 38 \, a b c^{4} d + 57 \, a^{2} c^{3} d^{2}\right )} \sqrt {d} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (1155 \, b^{2} d^{5} x^{8} + 180 \, b^{2} c^{4} d - 760 \, a b c^{3} d^{2} + 1140 \, a^{2} c^{2} d^{3} + 77 \, {\left (21 \, b^{2} c d^{4} + 38 \, a b d^{5}\right )} x^{6} + 7 \, {\left (12 \, b^{2} c^{2} d^{3} + 646 \, a b c d^{4} + 285 \, a^{2} d^{5}\right )} x^{4} - 3 \, {\left (36 \, b^{2} c^{3} d^{2} - 152 \, a b c^{2} d^{3} - 1235 \, a^{2} c d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {3}{2}}\right )}}{21945 \, d^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} d e x^{7} + {\left (b^{2} c + 2 \, a b d\right )} e x^{5} + a^{2} c e x + {\left (2 \, a b c + a^{2} d\right )} e x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}, x\right ) \]