\[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx \]
Optimal antiderivative \[ -\frac {2 e \left (6 b^{2} e^{2} g^{2}+c e g \left (-14 a e g -27 b d g +17 b e f \right )-2 c^{2} \left (42 d^{2} g^{2}-111 d e f g +64 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{315 c^{2} g^{4}}-\frac {2 e^{2} \left (-b e g -6 c d g +8 c e f \right ) \left (g x +f \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{63 c \,g^{4}}+\frac {2 \left (8 b^{3} e^{3} g^{3}+3 b c \,e^{2} g^{2} \left (-9 a e g -12 b d g +5 b e f \right )-c^{3} \left (-70 d^{3} g^{3}+336 d^{2} e f \,g^{2}-408 d \,e^{2} f^{2} g +152 e^{3} f^{3}\right )-3 c^{2} e g \left (6 a e g \left (-5 d g +2 e f \right )-b \left (21 d^{2} g^{2}-24 d e f g +8 e^{2} f^{2}\right )\right )\right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{315 c^{3} g^{4}}+\frac {2 \left (e x +d \right )^{3} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}{9 g}-\frac {\left (16 b^{4} e^{3} g^{4}+8 b^{2} c \,e^{2} g^{3} \left (-9 a e g -9 b d g +2 b e f \right )-2 c^{4} f \left (-105 d^{3} g^{3}+252 d^{2} e f \,g^{2}-216 d \,e^{2} f^{2} g +64 e^{3} f^{3}\right )+3 c^{2} e \,g^{2} \left (14 a^{2} e^{2} g^{2}-a b e g \left (-87 d g +19 e f \right )+b^{2} \left (42 d^{2} g^{2}-27 d e f g +7 e^{2} f^{2}\right )\right )-c^{3} g \left (6 a e g \left (63 d^{2} g^{2}-39 d e f g +10 e^{2} f^{2}\right )-b \left (-105 d^{3} g^{3}+189 d^{2} e f \,g^{2}-144 d \,e^{2} f^{2} g +40 e^{3} f^{3}\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}}}}{315 c^{4} g^{5} \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 (8 b^{3} e^{3} g^{3}+3 b c \,e^{2} g^{2} \left (-9 a e g -12 b d g +5 b e f \right )+2 c^{3} \left (-105 d^{3} g^{3}+252 d^{2} e f \,g^{2}-216 d \,e^{2} f^{2} g +64 e^{3} f^{3}\right )-3 c^{2} e g \left (6 a e g \left (-5 d g +2 e f \right )-b \left (21 d^{2} g^{2}-24 d e f g +8 e^{2} f^{2}\right )\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 )}}}{315 c^{4} g^{5} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^3*(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 (210 \, c^{5} d^{3} f^{2} g^{3} - 210 \, b c^{4} d^{3} f g^{4} - 105 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d^{3} g^{5} - {\left (128 \, c^{5} f^{5} - 104 \, b c^{4} f^{4} g - {\left (25 \, b^{2} c^{3} - 156 \, a c^{4}\right )} f^{3} g^{2} - 5 \, {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} f^{2} g^{3} - {\left (8 \, b^{4} c - 39 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} f g^{4} - {\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} g^{5}\right )} e^{3} + 9 \, {\left (48 \, c^{5} d f^{4} g - 40 \, b c^{4} d f^{3} g^{2} - 2 \, {\left (5 \, b^{2} c^{3} - 31 \, a c^{4}\right )} d f^{2} g^{3} - {\left (5 \, b^{3} c^{2} - 22 \, a b c^{3}\right )} d f g^{4} - {\left (8 \, b^{4} c - 41 \, a b^{2} c^{2} + 30 \, a^{2} c^{3}\right )} d g^{5}\right )} e^{2} - 63 \, {\left (8 \, c^{5} d^{2} f^{3} g^{2} - 7 \, b c^{4} d^{2} f^{2} g^{3} - 2 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d^{2} f g^{4} - {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} d^{2} g^{5}\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 (210 \, c^{5} d^{3} f g^{4} - 105 \, b c^{4} d^{3} g^{5} - {\left (128 \, c^{5} f^{4} g - 40 \, b c^{4} f^{3} g^{2} - 3 \, {\left (7 \, b^{2} c^{3} - 20 \, a c^{4}\right )} f^{2} g^{3} - {\left (16 \, b^{3} c^{2} - 57 \, a b c^{3}\right )} f g^{4} - 2 \, {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} g^{5}\right )} e^{3} + 9 \, {\left (48 \, c^{5} d f^{3} g^{2} - 16 \, b c^{4} d f^{2} g^{3} - {\left (9 \, b^{2} c^{3} - 26 \, a c^{4}\right )} d f g^{4} - {\left (8 \, b^{3} c^{2} - 29 \, a b c^{3}\right )} d g^{5}\right )} e^{2} - 63 \, {\left (8 \, c^{5} d^{2} f^{2} g^{3} - 3 \, b c^{4} d^{2} f g^{4} - 2 \, {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} d^{2} g^{5}\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 (105 \, c^{5} d^{3} g^{5} + {\left (35 \, c^{5} g^{5} x^{3} - 64 \, c^{5} f^{3} g^{2} + 12 \, b c^{4} f^{2} g^{3} + {\left (9 \, b^{2} c^{3} - 22 \, a c^{4}\right )} f g^{4} + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} g^{5} - 5 \, {\left (8 \, c^{5} f g^{4} - b c^{4} g^{5}\right )} x^{2} + {\left (48 \, c^{5} f^{2} g^{3} - 7 \, b c^{4} f g^{4} - 2 \, {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} g^{5}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c^{5} d g^{5} x^{2} + 24 \, c^{5} d f^{2} g^{3} - 5 \, b c^{4} d f g^{4} - 2 \, {\left (2 \, b^{2} c^{3} - 5 \, a c^{4}\right )} d g^{5} - 3 \, {\left (6 \, c^{5} d f g^{4} - b c^{4} d g^{5}\right )} x\right )} e^{2} + 63 \, {\left (3 \, c^{5} d^{2} g^{5} x - 4 \, c^{5} d^{2} f g^{4} + b c^{4} d^{2} g^{5}\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{945 \, c^{5} g^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + b x + a}}{\sqrt {g x + f}}, x\right ) \]