\[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ \frac {2 a \left (-2 a g +5 b d \right ) \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {6 a e x \sqrt {b \,x^{3}+a}}{55 b}+\frac {6 a f \,x^{2} \sqrt {b \,x^{3}+a}}{91 b}+\frac {2 a g \,x^{3} \sqrt {b \,x^{3}+a}}{45 b}+\frac {2 x \left (3003 g \,x^{5}+3465 f \,x^{4}+4095 e \,x^{3}+5005 d \,x^{2}+6435 c x \right ) \sqrt {b \,x^{3}+a}}{45045}+\frac {6 a \left (-4 a f +13 b c \right ) \sqrt {b \,x^{3}+a}}{91 b^{\frac {5}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {3 \,3^{\frac {1}{4}} a^{\frac {4}{3}} \left (-4 a f +13 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}}}}{91 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 {2 \,3^{\frac {3}{4}} a^{\frac {4}{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 (182 a^{\frac {2}{3}} b^{\frac {1}{3}} e +55 \left (-4 a f +13 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}}}}{5005 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*(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
\[ -\frac {2 \, {\left (4914 \, a^{2} \sqrt {b} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 1485 \, {\left (13 \, a b c - 4 \, a^{2} f\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (3003 \, b^{2} g x^{6} + 3465 \, b^{2} f x^{5} + 1001 \, {\left (5 \, b^{2} d + a b g\right )} x^{3} + 5005 \, a b d - 2002 \, a^{2} g + 495 \, {\left (13 \, b^{2} c + 3 \, a b f\right )} x^{2} + 819 \, {\left (5 \, b^{2} x^{4} + 3 \, a b x\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{45045 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (g x^{5} + f x^{4} + e x^{3} + d x^{2} + c x\right )} \sqrt {b x^{3} + a}, x\right ) \]