\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (-10 a d +3 b c \right ) \left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{55 d^{2} e}+\frac {2 b^{2} \left (e x \right )^{\frac {9}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 d \,e^{3}}+\frac {2 \left (11 a^{2} d^{2}+b c \left (-10 a d +3 b c \right )\right ) \left (e x \right )^{\frac {5}{2}} \sqrt {d \,x^{2}+c}}{77 d^{2} e}+\frac {4 c \left (11 a^{2} d^{2}+b c \left (-10 a d +3 b c \right )\right ) e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{231 d^{3}}-\frac {2 c^{\frac {7}{4}} \left (11 a^{2} d^{2}+b c \left (-10 a d +3 b c \right )\right ) e^{\frac {3}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {13}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((e*x)^(3/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (10 \, {\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 11 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (77 \, b^{2} d^{4} x^{6} + 30 \, b^{2} c^{3} d - 100 \, a b c^{2} d^{2} + 110 \, a^{2} c d^{3} + 14 \, {\left (b^{2} c d^{3} + 15 \, a b d^{4}\right )} x^{4} - 3 \, {\left (6 \, b^{2} c^{2} d^{2} - 20 \, a b c d^{3} - 55 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {3}{2}}\right )}}{1155 \, d^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} e x^{5} + 2 \, a b e x^{3} + a^{2} e x\right )} \sqrt {d x^{2} + c} \sqrt {e x}, x\right ) \]