\[ \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \sqrt {c \,x^{2}+a}}{5 a e \left (e x \right )^{\frac {5}{2}}}-\frac {2 B \sqrt {c \,x^{2}+a}}{3 a \,e^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {6 A c \sqrt {c \,x^{2}+a}}{5 a^{2} e^{3} \sqrt {e x}}-\frac {6 A \,c^{\frac {3}{2}} x \sqrt {c \,x^{2}+a}}{5 a^{2} e^{3} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {6 A \,c^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {c^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (5 B \sqrt {a}+9 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 \, B a \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9 \, A c^{\frac {3}{2}} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (9 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{15 \, a^{2} x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c e^{4} x^{6} + a e^{4} x^{4}}, x\right ) \]