\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (e x \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{13 c}-\frac {2 a e \left (77 B x +39 A \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{3003 c}+\frac {2 A e \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x}}{11 c}-\frac {8 a^{3} B \,e^{2} x \sqrt {c \,x^{2}+a}}{65 c^{\frac {3}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {4 a^{2} e \left (77 B x +65 A \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{5005 c}+\frac {8 a^{\frac {13}{4}} B \,e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{65 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {4 a^{\frac {11}{4}} e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (77 B \sqrt {a}+65 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5005 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(3/2)*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (780 \, A a^{3} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 924 \, B a^{3} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (1155 \, B c^{3} x^{5} + 1365 \, A c^{3} x^{4} + 1925 \, B a c^{2} x^{3} + 2535 \, A a c^{2} x^{2} + 308 \, B a^{2} c x + 780 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{15015 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c e x^{4} + A c e x^{3} + B a e x^{2} + A a e x\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]