\[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx \]
Optimal antiderivative \[ \frac {30 d \left (667 a \,e^{2}-58 b d e +16 c \,d^{2}\right ) x \left (e \,x^{3}+d \right )^{\frac {3}{2}}}{124729 e^{2}}+\frac {2 \left (667 a \,e^{2}-58 b d e +16 c \,d^{2}\right ) x \left (e \,x^{3}+d \right )^{\frac {5}{2}}}{11339 e^{2}}-\frac {2 \left (-29 b e +8 c d \right ) x \left (e \,x^{3}+d \right )^{\frac {7}{2}}}{667 e^{2}}+\frac {2 c \,x^{4} \left (e \,x^{3}+d \right )^{\frac {7}{2}}}{29 e}+\frac {54 d^{2} \left (667 a \,e^{2}-58 b d e +16 c \,d^{2}\right ) x \sqrt {e \,x^{3}+d}}{124729 e^{2}}+\frac {54 \,3^{\frac {3}{4}} d^{3} \left (667 a \,e^{2}-58 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{124729 e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((e*x^3+d)^(5/2)*(c*x^6+b*x^3+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{124729} \, {\left (81 \, {\left (16 \, c d^{5} - 58 \, b d^{4} e + 667 \, a d^{3} e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (648 \, c d^{4} x e - 11 \, {\left (391 \, c x^{13} + 493 \, b x^{10} + 667 \, a x^{7}\right )} e^{5} - {\left (11407 \, c d x^{10} + 15631 \, b d x^{7} + 24679 \, a d x^{4}\right )} e^{4} - {\left (8591 \, c d^{2} x^{7} + 14123 \, b d^{2} x^{4} + 35351 \, a d^{2} x\right )} e^{3} - 81 \, {\left (5 \, c d^{3} x^{4} + 29 \, b d^{3} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c e^{2} x^{12} + {\left (2 \, c d e + b e^{2}\right )} x^{9} + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{6} + {\left (b d^{2} + 2 \, a d e\right )} x^{3} + a d^{2}\right )} \sqrt {e x^{3} + d}, x\right ) \]