\[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (36465 g \,x^{5}+40755 f \,x^{4}+46189 e \,x^{3}+53295 d \,x^{2}+62985 c x \right )}{692835}+\frac {2 a^{2} e \sqrt {b \,x^{3}+a}}{15 b}+\frac {54 a^{2} f x \sqrt {b \,x^{3}+a}}{935 b}+\frac {54 a^{2} g \,x^{2} \sqrt {b \,x^{3}+a}}{1729 b}+\frac {2 a \left (176715 g \,x^{5}+233415 f \,x^{4}+323323 e \,x^{3}+479655 d \,x^{2}+793611 c x \right ) \sqrt {b \,x^{3}+a}}{4849845}+\frac {54 a^{2} \left (-4 a g +19 b d \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 g +19 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}}}}{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^{2} \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 (-2 a f +17 b c \right )-935 a^{\frac {1}{3}} \left (-4 a g +19 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}}}}{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((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
\[ \frac {2 \, {\left (140049 \, {\left (17 \, a^{2} b c - 2 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 75735 \, {\left (19 \, a^{2} b d - 4 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (255255 \, b^{3} g x^{8} + 285285 \, b^{3} f x^{7} + 19635 \, {\left (19 \, b^{3} d + 22 \, a b^{2} g\right )} x^{5} + 25935 \, {\left (17 \, b^{3} c + 20 \, a b^{2} f\right )} x^{4} + 2805 \, {\left (304 \, a b^{2} d + 27 \, a^{2} b g\right )} x^{2} + 5187 \, {\left (238 \, a b^{2} c + 27 \, a^{2} b f\right )} x + 323323 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{4849845 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}, x\right ) \]