\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{21 d \,e^{2} \left (e \,x^{3}+d \right )^{\frac {7}{2}}}-\frac {2 \left (-19 a \,e^{2}-2 b d e +23 c \,d^{2}\right ) x}{315 d^{2} e^{2} \left (e \,x^{3}+d \right )^{\frac {5}{2}}}+\frac {2 \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right ) x}{2835 d^{3} e^{2} \left (e \,x^{3}+d \right )^{\frac {3}{2}}}+\frac {2 \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right ) x}{1215 d^{4} e^{2} \sqrt {e \,x^{3}+d}}+\frac {2 \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{3645 d^{4} e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (7 \, {\left (247 \, a x^{12} e^{6} + 16 \, c d^{6} + 26 \, {\left (b d x^{12} + 38 \, a d x^{9}\right )} e^{5} + 2 \, {\left (8 \, c d^{2} x^{12} + 52 \, b d^{2} x^{9} + 741 \, a d^{2} x^{6}\right )} e^{4} + 4 \, {\left (16 \, c d^{3} x^{9} + 39 \, b d^{3} x^{6} + 247 \, a d^{3} x^{3}\right )} e^{3} + {\left (96 \, c d^{4} x^{6} + 104 \, b d^{4} x^{3} + 247 \, a d^{4}\right )} e^{2} + 2 \, {\left (32 \, c d^{5} x^{3} + 13 \, b d^{5}\right )} e\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) + {\left (1729 \, a x^{10} e^{6} - 56 \, c d^{5} x e + 26 \, {\left (7 \, b d x^{10} + 228 \, a d x^{7}\right )} e^{5} + 2 \, {\left (56 \, c d^{2} x^{10} + 312 \, b d^{2} x^{7} + 3591 \, a d^{2} x^{4}\right )} e^{4} + 4 \, {\left (96 \, c d^{3} x^{7} + 189 \, b d^{3} x^{4} + 847 \, a d^{3} x\right )} e^{3} - 7 \, {\left (27 \, c d^{4} x^{4} + 13 \, b d^{4} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )}}{8505 \, {\left (d^{4} x^{12} e^{7} + 4 \, d^{5} x^{9} e^{6} + 6 \, d^{6} x^{6} e^{5} + 4 \, d^{7} x^{3} e^{4} + d^{8} e^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d}}{e^{5} x^{15} + 5 \, d e^{4} x^{12} + 10 \, d^{2} e^{3} x^{9} + 10 \, d^{3} e^{2} x^{6} + 5 \, d^{4} e x^{3} + d^{5}}, x\right ) \]