\[ \int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {x}}{7 c}-\frac {2 \left (4 b^{2} B -7 A b c +5 a B c +3 c \left (-7 A c +4 b B \right ) x \right ) \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{105 c^{2}}-\frac {2 \left (5 a b B c -2 \left (-3 a c +b^{2}\right ) \left (-7 A c +4 b B \right )\right ) \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{105 c^{\frac {5}{2}} \left (\sqrt {a}+x \sqrt {c}\right )}+\frac {2 a^{\frac {1}{4}} \left (5 a b B c -2 \left (-3 a c +b^{2}\right ) \left (-7 A c +4 b B \right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{2}+b x +a}}-\frac {a^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \left (5 a b B c -2 \left (-3 a c +b^{2}\right ) \left (-7 A c +4 b B \right )-\left (-7 A b c -10 a B c +4 b^{2} B \right ) \sqrt {a}\, \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((B*x+A)*x^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (8 \, B b^{4} + 3 \, {\left (10 \, B a^{2} + 21 \, A a b\right )} c^{2} - {\left (41 \, B a b^{2} + 14 \, A b^{3}\right )} c\right )} \sqrt {c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (8 \, B b^{3} c + 42 \, A a c^{3} - {\left (29 \, B a b + 14 \, A b^{2}\right )} c^{2}\right )} \sqrt {c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (15 \, B c^{4} x^{2} - 4 \, B b^{2} c^{2} + {\left (10 \, B a + 7 \, A b\right )} c^{3} + 3 \, {\left (B b c^{3} + 7 \, A c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{315 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} {\left (B x + A\right )} \sqrt {x}, x\right ) \]