\[ \int \frac {(d+e x) \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-3 c e g x -b e g -5 c d g +4 c e f \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{15 c \,g^{2}}-\frac {\left (2 b^{2} e \,g^{2}-2 c^{2} f \left (-5 d g +4 e f \right )+c g \left (-6 a e g -5 b d g +3 b e f \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {g x +f}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{2} g^{3} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (b e g -10 c d g +8 c e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 c^{2} g^{3} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)*(c*x^2+b*x+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (10 \, c^{3} d f^{2} g - 10 \, b c^{2} d f g^{2} - 5 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d g^{3} - {\left (8 \, c^{3} f^{3} - 7 \, b c^{2} f^{2} g - 2 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} - {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (10 \, c^{3} d f g^{2} - 5 \, b c^{2} d g^{3} - {\left (8 \, c^{3} f^{2} g - 3 \, b c^{2} f g^{2} - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} g^{3}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (5 \, c^{3} d g^{3} + {\left (3 \, c^{3} g^{3} x - 4 \, c^{3} f g^{2} + b c^{2} g^{3}\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{3} g^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}{\sqrt {g x + f}}, x\right ) \]