\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3 a A +\left (A b +5 B a \right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{15 a \,x^{\frac {5}{2}}}+\frac {2 \left (-6 a A c +2 A \,b^{2}-5 a b B \right ) \sqrt {c \,x^{2}+b x +a}}{15 a^{2} \sqrt {x}}+\frac {2 \left (5 a b B -2 A \left (-3 a c +b^{2}\right )\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{15 a^{2} \left (\sqrt {a}+x \sqrt {c}\right )}-\frac {2 c^{\frac {1}{4}} \left (5 a b B -2 A \left (-3 a c +b^{2}\right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \sqrt {c \,x^{2}+b x +a}}-\frac {c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \left (b +2 \sqrt {a}\, \sqrt {c}\right ) \left (2 A b -5 B a -3 A \sqrt {a}\, \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3} - 3 \, {\left (10 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (6 \, A a c^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} c\right )} \sqrt {c} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (3 \, A a^{2} c + {\left (5 \, B a^{2} + A a b\right )} c x + {\left (6 \, A a c^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} c\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{45 \, a^{2} c x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {7}{2}}}, x\right ) \]