\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{15 d \,e^{2} \left (e \,x^{3}+d \right )^{\frac {5}{2}}}-\frac {2 \left (-13 a \,e^{2}-2 b d e +17 c \,d^{2}\right ) x}{135 d^{2} e^{2} \left (e \,x^{3}+d \right )^{\frac {3}{2}}}+\frac {2 \left (91 a \,e^{2}+14 b d e +16 c \,d^{2}\right ) x}{405 d^{3} e^{2} \sqrt {e \,x^{3}+d}}+\frac {2 \left (91 a \,e^{2}+14 b d e +16 c \,d^{2}\right ) \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{1215 d^{3} e^{\frac {7}{3}} \sqrt {e \,x^{3}+d}\, \sqrt {\frac {d^{\frac {1}{3}} \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{\left (e^{\frac {1}{3}} x +d^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(7/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {7}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{3}, \frac {7}{2} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \Gamma \left (\frac {10}{3}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________