\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x^{3} \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (156009 g \,x^{5}+169575 f \,x^{4}+185725 e \,x^{3}+205275 d \,x^{2}+229425 c x \right )}{3900225}-\frac {4 a^{3} e \sqrt {b \,x^{3}+a}}{105 b^{2}}+\frac {54 a^{2} \left (-8 a f +23 b c \right ) x \sqrt {b \,x^{3}+a}}{21505 b^{2}}+\frac {54 a^{2} \left (-2 a g +5 b d \right ) x^{2} \sqrt {b \,x^{3}+a}}{8645 b^{2}}+\frac {2 a^{2} e \,x^{3} \sqrt {b \,x^{3}+a}}{105 b}+\frac {54 a^{2} f \,x^{4} \sqrt {b \,x^{3}+a}}{4301 b}+\frac {54 a^{2} g \,x^{5} \sqrt {b \,x^{3}+a}}{6175 b}+\frac {2 a \,x^{3} \left (3522519 g \,x^{5}+4279275 f \,x^{4}+5311735 e \,x^{3}+6774075 d \,x^{2}+8947575 c x \right ) \sqrt {b \,x^{3}+a}}{185910725}-\frac {216 a^{3} \left (-2 a g +5 b d \right ) \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 {108 \,3^{\frac {1}{4}} a^{\frac {10}{3}} \left (-2 a g +5 b d \right ) \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}}}}-\frac {36 \,3^{\frac {3}{4}} a^{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 ) \left (1729 b^{\frac {1}{3}} \left (-8 a f +23 b c \right )-8602 a^{\frac {1}{3}} \left (-2 a g +5 b d \right ) \left (1-\sqrt {3}\right )\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}}}}{37182145 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^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (1400490 \, {\left (23 \, a^{3} b c - 8 \, a^{4} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 6967620 \, {\left (5 \, a^{3} b d - 2 \, a^{4} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (22309287 \, b^{4} g x^{11} + 24249225 \, b^{4} f x^{10} + 1174173 \, {\left (25 \, b^{4} d + 28 \, a b^{3} g\right )} x^{8} + 1426425 \, {\left (23 \, b^{4} c + 26 \, a b^{3} f\right )} x^{7} + 90321 \, {\left (550 \, a b^{3} d + 27 \, a^{2} b^{2} g\right )} x^{5} + 129675 \, {\left (460 \, a b^{3} c + 27 \, a^{2} b^{2} f\right )} x^{4} + 1741905 \, {\left (5 \, a^{2} b^{2} d - 2 \, a^{3} b g\right )} x^{2} + 700245 \, {\left (23 \, a^{2} b^{2} c - 8 \, a^{3} b f\right )} x + 5311735 \, {\left (5 \, b^{4} x^{9} + 8 \, a b^{3} x^{6} + a^{2} b^{2} x^{3} - 2 \, a^{3} b\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{557732175 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b g x^{10} + b f x^{9} + b e x^{8} + {\left (b d + a g\right )} x^{7} + a e x^{5} + {\left (b c + a f\right )} x^{6} + a d x^{4} + a c x^{3}\right )} \sqrt {b x^{3} + a}, x\right ) \]