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