\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx \]
Optimal antiderivative \[ \frac {4 e \left (7 a \,e^{2} g^{2}+c \left (42 d^{2} g^{2}-111 d e f g +64 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+a}}{315 c \,g^{4}}-\frac {4 e^{2} \left (-3 d g +4 e f \right ) \left (g x +f \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+a}}{63 g^{4}}-\frac {4 \left (9 a \,e^{2} g^{2} \left (-5 d g +2 e f \right )+c \left (-35 d^{3} g^{3}+168 d^{2} e f \,g^{2}-204 d \,e^{2} f^{2} g +76 e^{3} f^{3}\right )\right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{315 c \,g^{4}}+\frac {2 \left (e x +d \right )^{3} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{9 g}+\frac {4 \left (21 a^{2} e^{3} g^{4}-3 a c e \,g^{2} \left (63 d^{2} g^{2}-39 d e f g +10 e^{2} f^{2}\right )-c^{2} f \left (-105 d^{3} g^{3}+252 d^{2} e f \,g^{2}-216 d \,e^{2} f^{2} g +64 e^{3} f^{3}\right )\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {g x +f}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{315 c^{\frac {3}{2}} g^{5} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}-\frac {4 \left (a \,g^{2}+c \,f^{2}\right ) \left (9 a \,e^{2} g^{2} \left (-5 d g +2 e f \right )-c \left (-105 d^{3} g^{3}+252 d^{2} e f \,g^{2}-216 d \,e^{2} f^{2} g +64 e^{3} f^{3}\right )\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}{315 c^{\frac {3}{2}} g^{5} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x+d)^3*(c*x^2+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
\[ \frac {2 \, {\left (2 \, {\left (105 \, c^{2} d^{3} f^{2} g^{3} + 315 \, a c d^{3} g^{5} - 2 \, {\left (32 \, c^{2} f^{5} + 39 \, a c f^{3} g^{2} - 6 \, a^{2} f g^{4}\right )} e^{3} + 9 \, {\left (24 \, c^{2} d f^{4} g + 31 \, a c d f^{2} g^{3} - 15 \, a^{2} d g^{5}\right )} e^{2} - 126 \, {\left (2 \, c^{2} d^{2} f^{3} g^{2} + 3 \, a c d^{2} f g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (105 \, c^{2} d^{3} f g^{4} - {\left (64 \, c^{2} f^{4} g + 30 \, a c f^{2} g^{3} - 21 \, a^{2} g^{5}\right )} e^{3} + 9 \, {\left (24 \, c^{2} d f^{3} g^{2} + 13 \, a c d f g^{4}\right )} e^{2} - 63 \, {\left (4 \, c^{2} d^{2} f^{2} g^{3} + 3 \, a c d^{2} g^{5}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (105 \, c^{2} d^{3} g^{5} + {\left (35 \, c^{2} g^{5} x^{3} - 40 \, c^{2} f g^{4} x^{2} - 64 \, c^{2} f^{3} g^{2} - 22 \, a c f g^{4} + 2 \, {\left (24 \, c^{2} f^{2} g^{3} + 7 \, a c g^{5}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c^{2} d g^{5} x^{2} - 18 \, c^{2} d f g^{4} x + 24 \, c^{2} d f^{2} g^{3} + 10 \, a c d g^{5}\right )} e^{2} + 63 \, {\left (3 \, c^{2} d^{2} g^{5} x - 4 \, c^{2} d^{2} f g^{4}\right )} e\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + a}}{\sqrt {g x + f}}, x\right ) \]