\[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 x^{8} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{25}-\frac {108 a^{3} x^{2} \sqrt {b \,x^{3}+a}}{8645 b^{2}}+\frac {54 a^{2} x^{5} \sqrt {b \,x^{3}+a}}{6175 b}+\frac {18 a \,x^{8} \sqrt {b \,x^{3}+a}}{475}+\frac {432 a^{4} \sqrt {b \,x^{3}+a}}{8645 b^{\frac {8}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {144 \,3^{\frac {3}{4}} a^{\frac {13}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{8645 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {216 \,3^{\frac {1}{4}} a^{\frac {13}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{8645 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^7*(b*x^3+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (1080 \, a^{4} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (1729 \, b^{4} x^{11} + 2548 \, a b^{3} x^{8} + 189 \, a^{2} b^{2} x^{5} - 270 \, a^{3} b x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{43225 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b x^{10} + a x^{7}\right )} \sqrt {b x^{3} + a}, x\right ) \]