\[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (391 a \,e^{2}-46 b d e +16 c \,d^{2}\right ) x \left (e \,x^{3}+d \right )^{\frac {3}{2}}}{4301 e^{2}}-\frac {2 \left (-23 b e +8 c d \right ) x \left (e \,x^{3}+d \right )^{\frac {5}{2}}}{391 e^{2}}+\frac {2 c \,x^{4} \left (e \,x^{3}+d \right )^{\frac {5}{2}}}{23 e}+\frac {18 d \left (391 a \,e^{2}-46 b d e +16 c \,d^{2}\right ) x \sqrt {e \,x^{3}+d}}{21505 e^{2}}+\frac {18 \,3^{\frac {3}{4}} d^{2} \left (391 a \,e^{2}-46 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{21505 e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{21505} \, {\left (27 \, {\left (16 \, c d^{4} - 46 \, b d^{3} e + 391 \, a d^{2} e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (216 \, c d^{3} x e - 5 \, {\left (187 \, c x^{10} + 253 \, b x^{7} + 391 \, a x^{4}\right )} e^{4} - 2 \, {\left (715 \, c d x^{7} + 1150 \, b d x^{4} + 2737 \, a d x\right )} e^{3} - 27 \, {\left (5 \, c d^{2} x^{4} + 23 \, b d^{2} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c e x^{9} + {\left (c d + b e\right )} x^{6} + {\left (b d + a e\right )} x^{3} + a d\right )} \sqrt {e x^{3} + d}, x\right ) \]