\[ \int \frac {(e x)^{7/2} (A+B x)}{\sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {14 a B \,e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+a}}{45 c^{2}}+\frac {2 A e \left (e x \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+a}}{7 c}+\frac {2 B \left (e x \right )^{\frac {7}{2}} \sqrt {c \,x^{2}+a}}{9 c}+\frac {14 a^{2} B \,e^{4} x \sqrt {c \,x^{2}+a}}{15 c^{\frac {5}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {10 a A \,e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{21 c^{2}}-\frac {14 a^{\frac {9}{4}} B \,e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {a^{\frac {7}{4}} e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (49 B \sqrt {a}+25 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
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
\[ \frac {2 \, {\left (75 \, A a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 147 \, B a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (35 \, B c^{2} x^{3} + 45 \, A c^{2} x^{2} - 49 \, B a c x - 75 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {7}{2}}\right )}}{315 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt {e x}}{\sqrt {c x^{2} + a}}, x\right ) \]