\[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {e \left (e x \right )^{\frac {5}{2}} \left (B x +A \right )}{c \sqrt {c \,x^{2}+a}}+\frac {7 B \,e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+a}}{5 c^{2}}-\frac {21 a B \,e^{4} x \sqrt {c \,x^{2}+a}}{5 c^{\frac {5}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {5 A \,e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{3 c^{2}}+\frac {21 a^{\frac {5}{4}} B \,e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {a^{\frac {3}{4}} e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (63 B \sqrt {a}+25 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{30 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {25 \, {\left (A a c x^{2} + A a^{2}\right )} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 63 \, {\left (B a c x^{2} + B a^{2}\right )} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (6 \, B c^{2} x^{3} + 10 \, A c^{2} x^{2} + 21 \, B a c x + 25 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {7}{2}}}{15 \, {\left (c^{4} x^{2} + a c^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]