\[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-17 b e +8 c d \right ) x \left (e \,x^{3}+d \right )^{\frac {3}{2}}}{187 e^{2}}+\frac {2 c \,x^{4} \left (e \,x^{3}+d \right )^{\frac {3}{2}}}{17 e}+\frac {2 \left (187 a \,e^{2}-34 b d e +16 c \,d^{2}\right ) x \sqrt {e \,x^{3}+d}}{935 e^{2}}+\frac {2 \,3^{\frac {3}{4}} d \left (187 a \,e^{2}-34 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{935 e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{935} \, {\left (3 \, {\left (16 \, c d^{3} - 34 \, b d^{2} e + 187 \, a d e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (24 \, c d^{2} x e - {\left (55 \, c x^{7} + 85 \, b x^{4} + 187 \, a x\right )} e^{3} - 3 \, {\left (5 \, c d x^{4} + 17 \, b d x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d}, x\right ) \]