\[ \int x \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (20995 g \,x^{5}+23205 f \,x^{4}+25935 e \,x^{3}+29393 d \,x^{2}+33915 c x \right )}{440895}+\frac {2 a^{2} \left (-2 a g +7 b d \right ) \sqrt {b \,x^{3}+a}}{105 b^{2}}+\frac {54 a^{2} e x \sqrt {b \,x^{3}+a}}{935 b}+\frac {54 a^{2} f \,x^{2} \sqrt {b \,x^{3}+a}}{1729 b}+\frac {2 a^{2} g \,x^{3} \sqrt {b \,x^{3}+a}}{105 b}+\frac {2 a x \left (138567 g \,x^{5}+176715 f \,x^{4}+233415 e \,x^{3}+323323 d \,x^{2}+479655 c x \right ) \sqrt {b \,x^{3}+a}}{4849845}+\frac {54 a^{2} \left (-4 a f +19 b c \right ) \sqrt {b \,x^{3}+a}}{1729 b^{\frac {5}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {27 \,3^{\frac {1}{4}} a^{\frac {7}{3}} \left (-4 a f +19 b c \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}}}}{1729 b^{\frac {5}{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 {18 \,3^{\frac {3}{4}} a^{\frac {7}{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 (3458 a^{\frac {2}{3}} b^{\frac {1}{3}} e +935 \left (-4 a f +19 b c \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}}}}{1616615 b^{\frac {5}{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*(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
\[ -\frac {2 \, {\left (280098 \, a^{3} \sqrt {b} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 75735 \, {\left (19 \, a^{2} b c - 4 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (230945 \, b^{3} g x^{9} + 255255 \, b^{3} f x^{8} + 46189 \, {\left (7 \, b^{3} d + 8 \, a b^{2} g\right )} x^{6} + 19635 \, {\left (19 \, b^{3} c + 22 \, a b^{2} f\right )} x^{5} + 323323 \, a^{2} b d - 92378 \, a^{3} g + 46189 \, {\left (14 \, a b^{2} d + a^{2} b g\right )} x^{3} + 2805 \, {\left (304 \, a b^{2} c + 27 \, a^{2} b f\right )} x^{2} + 5187 \, {\left (55 \, b^{3} x^{7} + 100 \, a b^{2} x^{4} + 27 \, a^{2} b x\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{4849845 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b g x^{8} + b f x^{7} + b e x^{6} + {\left (b d + a g\right )} x^{5} + a e x^{3} + {\left (b c + a f\right )} x^{4} + a d x^{2} + a c x\right )} \sqrt {b x^{3} + a}, x\right ) \]