\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{3 d \,e^{2} \sqrt {e \,x^{3}+d}}+\frac {2 c x \sqrt {e \,x^{3}+d}}{5 e^{2}}-\frac {2 \left (16 c \,d^{2}-5 e \left (a e +2 b d \right )\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{45 d \,e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (5 \, a x^{3} e^{3} - 16 \, c d^{3} + 5 \, {\left (2 \, b d x^{3} + a d\right )} e^{2} - 2 \, {\left (8 \, c d^{2} x^{3} - 5 \, b d^{2}\right )} e\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) + {\left (8 \, c d^{2} x e + 5 \, a x e^{3} + {\left (3 \, c d x^{4} - 5 \, b d x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )}}{15 \, {\left (d x^{3} e^{4} + d^{2} e^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d}}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}, x\right ) \]