\[ \int \frac {d+e x+f x^2}{(g+h x) \sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e h +f g \right ) \arctanh \left (\frac {x \sqrt {c}}{\sqrt {c \,x^{2}+a}}\right )}{h^{2} \sqrt {c}}-\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \arctanh \left (\frac {-c g x +a h}{\sqrt {a \,h^{2}+c \,g^{2}}\, \sqrt {c \,x^{2}+a}}\right )}{h^{2} \sqrt {a \,h^{2}+c \,g^{2}}}+\frac {f \sqrt {c \,x^{2}+a}}{c h} \]
command
integrate((f*x^2+e*x+d)/(h*x+g)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {{\left (c f g^{3} + a f g h^{2} - {\left (c g^{2} h + a h^{3}\right )} e\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (c f g^{2} + c d h^{2} - c g h e\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} + 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) - 2 \, {\left (c f g^{2} h + a f h^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} g^{2} h^{2} + a c h^{4}\right )}}, -\frac {2 \, {\left (c f g^{2} + c d h^{2} - c g h e\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) + {\left (c f g^{3} + a f g h^{2} - {\left (c g^{2} h + a h^{3}\right )} e\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (c f g^{2} h + a f h^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} g^{2} h^{2} + a c h^{4}\right )}}, \frac {2 \, {\left (c f g^{3} + a f g h^{2} - {\left (c g^{2} h + a h^{3}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c f g^{2} + c d h^{2} - c g h e\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} + 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) + 2 \, {\left (c f g^{2} h + a f h^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} g^{2} h^{2} + a c h^{4}\right )}}, -\frac {{\left (c f g^{2} + c d h^{2} - c g h e\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) - {\left (c f g^{3} + a f g h^{2} - {\left (c g^{2} h + a h^{3}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c f g^{2} h + a f h^{3}\right )} \sqrt {c x^{2} + a}}{c^{2} g^{2} h^{2} + a c h^{4}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]