\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (221 a^{2} d^{2}+3 b c \left (-34 a d +7 b c \right )\right ) \left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{1989 d^{2} e}-\frac {2 b \left (-34 a d +7 b c \right ) \left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{221 d^{2} e}+\frac {2 b^{2} \left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{17 d \,e^{3}}+\frac {4 c \left (221 a^{2} d^{2}+3 b c \left (-34 a d +7 b c \right )\right ) \left (e x \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}}{3315 d^{2} e}+\frac {8 c^{2} \left (221 a^{2} d^{2}+3 b c \left (-34 a d +7 b c \right )\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{3315 d^{\frac {5}{2}} \left (\sqrt {c}+x \sqrt {d}\right )}-\frac {8 c^{\frac {9}{4}} \left (221 a^{2} d^{2}+3 b c \left (-34 a d +7 b c \right )\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 {e}\, \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{3315 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {11}{4}} \sqrt {d \,x^{2}+c}}+\frac {4 c^{\frac {9}{4}} \left (221 a^{2} d^{2}+3 b c \left (-34 a d +7 b c \right )\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 {e}\, \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{3315 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {11}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (21 \, b^{2} c^{4} - 102 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (585 \, b^{2} d^{4} x^{7} + 45 \, {\left (19 \, b^{2} c d^{3} + 34 \, a b d^{4}\right )} x^{5} + 5 \, {\left (12 \, b^{2} c^{2} d^{2} + 510 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{3} - {\left (84 \, b^{2} c^{3} d - 408 \, a b c^{2} d^{2} - 2431 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {1}{2}}\right )}}{9945 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} d x^{6} + {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c + {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}, x\right ) \]