\[ \int (e x)^{3/2} (A+B x) \sqrt {a+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (e x \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{9 c}+\frac {2 A e \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{7 c}-\frac {4 a^{2} B \,e^{2} x \sqrt {c \,x^{2}+a}}{15 c^{\frac {3}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {2 a e \left (7 B x +5 A \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{105 c}+\frac {4 a^{\frac {9}{4}} B \,e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {2 a^{\frac {7}{4}} e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (7 B \sqrt {a}+5 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(3/2)*(B*x+A)*(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (30 \, A a^{2} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 42 \, B a^{2} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (35 \, B c^{2} x^{3} + 45 \, A c^{2} x^{2} + 14 \, B a c x + 30 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{315 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B e x^{2} + A e x\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]