\[ \int \frac {(d+e x) \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 e \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 \left (3 d g +e f \right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {g x +f}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{3 g \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}+\frac {2 e \left (a \,g^{2}+c \,f^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a g}{-a g +f \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {c}}{g \sqrt {-a}+f \sqrt {c}}}}{3 c^{\frac {3}{2}} g \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x+d)*(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
\[ \frac {2 \, {\left (3 \, \sqrt {c x^{2} + a} \sqrt {g x + f} c g^{2} e + {\left (6 \, c d f g - {\left (c f^{2} + 3 \, a g^{2}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) - 3 \, {\left (3 \, c d g^{2} + c f g e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )\right )}}{9 \, c^{2} g^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e x + d\right )} \sqrt {g x + f}}{\sqrt {c x^{2} + a}}, x\right ) \]