\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 e^{2} \left (11 d g +e f \right ) \left (g x +f \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+a}}{35 c \,g^{2}}-\frac {2 e \left (25 a \,e^{2} g^{2}+c \left (-90 d^{2} g^{2}+12 d e f g +7 e^{2} f^{2}\right )\right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{105 c^{2} g^{2}}+\frac {2 e \left (e x +d \right )^{2} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{7 c}+\frac {2 \left (a \,e^{2} g^{2} \left (189 d g +19 e f \right )-c \left (105 d^{3} g^{3}+105 d^{2} e f \,g^{2}-42 d \,e^{2} f^{2} g +8 e^{3} f^{3}\right )\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {g x +f}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{105 c^{\frac {3}{2}} g^{3} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}-\frac {2 e \left (a \,g^{2}+c \,f^{2}\right ) \left (25 a \,e^{2} g^{2}-c \left (105 d^{2} g^{2}-42 d e f g +8 e^{2} f^{2}\right )\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}{105 c^{\frac {5}{2}} g^{3} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (210 \, c^{2} d^{3} f g^{3} - {\left (8 \, c^{2} f^{4} - 13 \, a c f^{2} g^{2} - 75 \, a^{2} g^{4}\right )} e^{3} + 42 \, {\left (c^{2} d f^{3} g - 6 \, a c d f g^{3}\right )} e^{2} - 105 \, {\left (c^{2} d^{2} f^{2} g^{2} + 3 \, a c d^{2} g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) - 3 \, {\left (105 \, c^{2} d^{3} g^{4} + 105 \, c^{2} d^{2} f g^{3} e + {\left (8 \, c^{2} f^{3} g - 19 \, a c f g^{3}\right )} e^{3} - 21 \, {\left (2 \, c^{2} d f^{2} g^{2} + 9 \, a c d g^{4}\right )} e^{2}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (105 \, c^{2} d^{2} g^{4} e + {\left (15 \, c^{2} g^{4} x^{2} + 3 \, c^{2} f g^{3} x - 4 \, c^{2} f^{2} g^{2} - 25 \, a c g^{4}\right )} e^{3} + 21 \, {\left (3 \, c^{2} d g^{4} x + c^{2} d f g^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{3} g^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {g x + f}}{\sqrt {c x^{2} + a}}, x\right ) \]