\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (-4 b e g +7 c d g +c e f \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{15 c^{2} g}+\frac {2 e \left (e x +d \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{5 c}+\frac {\left (8 b^{2} e^{2} g^{2}-c e g \left (9 a e g +20 b d g +3 b e f \right )-c^{2} \left (-15 d^{2} g^{2}-10 d e f g +2 e^{2} f^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {g x +f}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{3} g^{2} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {4 e \left (2 b e g -5 c d g +c e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 g \sqrt {-4 a c +b^{2}}}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (g x +f \right )}{2 c f -g \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 c^{3} g^{2} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^2*(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
\[ \frac {2 \, {\left ({\left (30 \, c^{3} d^{2} f g^{2} - 15 \, b c^{2} d^{2} g^{3} + {\left (2 \, c^{3} f^{3} + 2 \, b c^{2} f^{2} g + {\left (7 \, b^{2} c - 12 \, a c^{2}\right )} f g^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} g^{3}\right )} e^{2} - 10 \, {\left (c^{3} d f^{2} g + 2 \, b c^{2} d f g^{2} - {\left (2 \, b^{2} c - 3 \, a c^{2}\right )} d g^{3}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) - 3 \, {\left (15 \, c^{3} d^{2} g^{3} - {\left (2 \, c^{3} f^{2} g + 3 \, b c^{2} f g^{2} - {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} g^{3}\right )} e^{2} + 10 \, {\left (c^{3} d f g^{2} - 2 \, b c^{2} d g^{3}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (10 \, c^{3} d g^{3} e + {\left (3 \, c^{3} g^{3} x + c^{3} f g^{2} - 4 \, b c^{2} g^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{4} g^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]