\[ \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx \]
Optimal antiderivative \[ -\frac {b \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2 f \sqrt {c}}-\frac {\sqrt {c \,x^{2}+b x +a}}{f}-\frac {\arctanh \left (\frac {b \sqrt {d}-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c d +a f -b \sqrt {d}\, \sqrt {f}}}\right ) \sqrt {c d +a f -b \sqrt {d}\, \sqrt {f}}}{2 f^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {b \sqrt {d}+2 a \sqrt {f}+x \left (2 c \sqrt {d}+b \sqrt {f}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c d +a f +b \sqrt {d}\, \sqrt {f}}}\right ) \sqrt {c d +a f +b \sqrt {d}\, \sqrt {f}}}{2 f^{\frac {3}{2}}} \]
command
integrate(x*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} + 2 \, b c d x + b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} - 2 \, b c d x - b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} + 2 \, b c d x + b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} - 2 \, b c d x - b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + b \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{2} + b x + a} c}{4 \, c f}, \frac {c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} + 2 \, b c d x + b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} - 2 \, b c d x - b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} + 2 \, b c d x + b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} - 2 \, b c d x - b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + 2 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 4 \, \sqrt {c x^{2} + b x + a} c}{4 \, c f}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]