\[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx \]
Optimal antiderivative \[ -\frac {14 a B \,e^{2} \left (e x \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{117 c^{2}}+\frac {2 A e \left (e x \right )^{\frac {5}{2}} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{11 c}+\frac {2 B \left (e x \right )^{\frac {7}{2}} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{13 c}-\frac {10 a A \,e^{3} \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{77 c^{2}}+\frac {28 a^{3} B \,e^{4} x \sqrt {c \,x^{2}+a}}{195 c^{\frac {5}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {2 a^{2} e^{3} \left (539 B x +325 A \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{15015 c^{2}}-\frac {28 a^{\frac {13}{4}} B \,e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{195 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {2 a^{\frac {11}{4}} e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (539 B \sqrt {a}+325 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15015 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(7/2)*(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
\[ \frac {2 \, {\left (1950 \, A a^{3} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3234 \, B a^{3} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3465 \, B c^{3} x^{5} + 4095 \, A c^{3} x^{4} + 770 \, B a c^{2} x^{3} + 1170 \, A a c^{2} x^{2} - 1078 \, B a^{2} c x - 1950 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {7}{2}}\right )}}{45045 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]