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