\[ \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {g x +f}}{7 c}-\frac {2 \left (4 b^{2} e \,g^{2}+c^{2} f \left (-7 d g +4 e f \right )-c g \left (-5 a e g +7 b d g +2 b e f \right )-3 c g \left (-4 b e g +7 c d g +c e f \right ) x \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{105 c^{2} g^{2}}+\frac {\left (\left (-4 b e g +7 c d g +c e f \right ) \left (8 c^{2} f^{2}-2 b^{2} g^{2}-3 c g \left (-2 a g +b f \right )\right )-5 c g \left (-b g +2 c f \right ) \left (7 c d f -e \left (a g +3 b f \right )\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {g x +f}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{105 c^{3} g^{3} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right ) \left (4 b^{2} e \,g^{2}-2 c^{2} f \left (-7 d g +4 e f \right )+c g \left (-10 a e g -7 b d g +b e f \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{105 c^{3} g^{3} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)*(g*x+f)^(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 (14 \, c^{4} d f^{3} g - 21 \, b c^{3} d f^{2} g^{2} - 21 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d f g^{3} + 7 \, {\left (2 \, b^{3} c - 9 \, a b c^{2}\right )} d g^{4} - {\left (8 \, c^{4} f^{4} - 9 \, b c^{3} f^{3} g - 2 \, {\left (2 \, b^{2} c^{2} - 11 \, a c^{3}\right )} f^{2} g^{2} - {\left (9 \, b^{3} c - 41 \, a b c^{2}\right )} f g^{3} + {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (14 \, c^{4} d f^{2} g^{2} - 14 \, b c^{3} d f g^{3} + 14 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} d g^{4} - {\left (8 \, c^{4} f^{3} g - 5 \, b c^{3} f^{2} g^{2} - {\left (5 \, b^{2} c^{2} - 16 \, a c^{3}\right )} f g^{3} + {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (21 \, c^{4} d g^{4} x + 7 \, c^{4} d f g^{3} + 7 \, b c^{3} d g^{4} + {\left (15 \, c^{4} g^{4} x^{2} - 4 \, c^{4} f^{2} g^{2} + 2 \, b c^{3} f g^{3} - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} g^{4} + 3 \, {\left (c^{4} f g^{3} + b c^{3} g^{4}\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{4} g^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \sqrt {g x + f}, x\right ) \]