\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{9 d \,e^{2} \left (e \,x^{3}+d \right )^{\frac {3}{2}}}-\frac {2 \left (-7 a \,e^{2}-2 b d e +11 c \,d^{2}\right ) x}{27 d^{2} e^{2} \sqrt {e \,x^{3}+d}}+\frac {2 \left (16 c \,d^{2}+e \left (7 a e +2 b d \right )\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{81 d^{2} e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (7 \, a x^{6} e^{4} + 16 \, c d^{4} + 2 \, {\left (b d x^{6} + 7 \, a d x^{3}\right )} e^{3} + {\left (16 \, c d^{2} x^{6} + 4 \, b d^{2} x^{3} + 7 \, a d^{2}\right )} e^{2} + 2 \, {\left (16 \, c d^{3} x^{3} + b d^{3}\right )} e\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) + {\left (7 \, a x^{4} e^{4} - 8 \, c d^{3} x e + 2 \, {\left (b d x^{4} + 5 \, a d x\right )} e^{3} - {\left (11 \, c d^{2} x^{4} + b d^{2} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )}}{27 \, {\left (d^{2} x^{6} e^{5} + 2 \, d^{3} x^{3} e^{4} + d^{4} e^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d}}{e^{3} x^{9} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{3} + d^{3}}, x\right ) \]