\[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-11 b e +8 c d \right ) x \sqrt {e \,x^{3}+d}}{55 e^{2}}+\frac {2 c \,x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (55 a \,e^{2}-22 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{165 e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{55} \, {\left ({\left (16 \, c d^{2} - 22 \, b d e + 55 \, a e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (8 \, c d x e - {\left (5 \, c x^{4} + 11 \, b x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {c x^{6} + b x^{3} + a}{\sqrt {e x^{3} + d}}, x\right ) \]