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