\[ \int \frac {x^3}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {1}{c f \sqrt {c \,x^{2}+a}}-\frac {e x}{a \,f^{2} \sqrt {c \,x^{2}+a}}+\frac {a f \left (c \,d^{2}+a \left (-d f +e^{2}\right )\right )+c e \left (c \,d^{2}+a \left (-2 d f +e^{2}\right )\right ) x}{a \,f^{2} \left (a c \,e^{2}+\left (-a f +c d \right )^{2}\right ) \sqrt {c \,x^{2}+a}}-\frac {\arctanh \left (\frac {\left (2 a f -c x \left (e -\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {c \,x^{2}+a}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f -e \sqrt {-4 d f +e^{2}}\right )}}\right ) \left (2 a d e f -\left (c \,d^{2}+a \left (-d f +e^{2}\right )\right ) \left (e -\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \left (a c \,e^{2}+\left (-a f +c d \right )^{2}\right ) \sqrt {-4 d f +e^{2}}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f -e \sqrt {-4 d f +e^{2}}\right )}}+\frac {\arctanh \left (\frac {\left (2 a f -c x \left (e +\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {c \,x^{2}+a}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f +e \sqrt {-4 d f +e^{2}}\right )}}\right ) \left (2 a d e f -\left (c \,d^{2}+a \left (-d f +e^{2}\right )\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \left (a c \,e^{2}+\left (-a f +c d \right )^{2}\right ) \sqrt {-4 d f +e^{2}}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f +e \sqrt {-4 d f +e^{2}}\right )}} \]
command
integrate(x^3/(c*x^2+a)^(3/2)/(f*x^2+e*x+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 \[ \text {Timed out} \]