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