\[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} e \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {6 a \left (-8 a f +17 b c \right ) x \sqrt {b \,x^{3}+a}}{935 b^{2}}+\frac {6 a \left (-10 a g +19 b d \right ) x^{2} \sqrt {b \,x^{3}+a}}{1729 b^{2}}+\frac {2 a e \,x^{3} \sqrt {b \,x^{3}+a}}{45 b}+\frac {6 a f \,x^{4} \sqrt {b \,x^{3}+a}}{187 b}+\frac {6 a g \,x^{5} \sqrt {b \,x^{3}+a}}{247 b}+\frac {2 x^{3} \left (36465 g \,x^{5}+40755 f \,x^{4}+46189 e \,x^{3}+53295 d \,x^{2}+62985 c x \right ) \sqrt {b \,x^{3}+a}}{692835}-\frac {24 a^{2} \left (-10 a g +19 b d \right ) \sqrt {b \,x^{3}+a}}{1729 b^{\frac {8}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {12 \,3^{\frac {1}{4}} a^{\frac {7}{3}} \left (-10 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 {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 {4 \,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 (-8 a f +17 b c \right )-1870 a^{\frac {1}{3}} \left (-10 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 {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*(g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (93366 \, {\left (17 \, a^{2} b c - 8 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 100980 \, {\left (19 \, a^{2} b d - 10 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (765765 \, b^{3} g x^{8} + 855855 \, b^{3} f x^{7} + 58905 \, {\left (19 \, b^{3} d + 3 \, a b^{2} g\right )} x^{5} + 77805 \, {\left (17 \, b^{3} c + 3 \, a b^{2} f\right )} x^{4} + 25245 \, {\left (19 \, a b^{2} d - 10 \, a^{2} b g\right )} x^{2} + 46683 \, {\left (17 \, a b^{2} c - 8 \, a^{2} b f\right )} x + 323323 \, {\left (3 \, b^{3} x^{6} + a b^{2} x^{3} - 2 \, a^{2} b\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{14549535 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (g x^{7} + f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )} \sqrt {b x^{3} + a}, x\right ) \]