\[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx \]
Optimal antiderivative \[ \frac {2 a \left (-2 a f +5 b c \right ) \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {6 a \left (-8 a g +17 b d \right ) x \sqrt {b \,x^{3}+a}}{935 b^{2}}+\frac {6 a e \,x^{2} \sqrt {b \,x^{3}+a}}{91 b}+\frac {2 a f \,x^{3} \sqrt {b \,x^{3}+a}}{45 b}+\frac {6 a g \,x^{4} \sqrt {b \,x^{3}+a}}{187 b}+\frac {2 x^{2} \left (6435 g \,x^{5}+7293 f \,x^{4}+8415 e \,x^{3}+9945 d \,x^{2}+12155 c x \right ) \sqrt {b \,x^{3}+a}}{109395}-\frac {24 a^{2} e \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 {12 \,3^{\frac {1}{4}} a^{\frac {7}{3}} e \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 {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 (1547 b d -728 a g -1870 a^{\frac {1}{3}} b^{\frac {2}{3}} e \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}}}}{85085 b^{\frac {7}{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^2*(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
\[ \frac {2 \, {\left (100980 \, a^{2} b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 4914 \, {\left (17 \, a^{2} b d - 8 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (45045 \, b^{3} g x^{7} + 51051 \, b^{3} f x^{6} + 4095 \, {\left (17 \, b^{3} d + 3 \, a b^{2} g\right )} x^{4} + 85085 \, a b^{2} c - 34034 \, a^{2} b f + 17017 \, {\left (5 \, b^{3} c + a b^{2} f\right )} x^{3} + 2457 \, {\left (17 \, a b^{2} d - 8 \, a^{2} b g\right )} x + 8415 \, {\left (7 \, b^{3} x^{5} + 3 \, a b^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{765765 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (g x^{6} + f x^{5} + e x^{4} + d x^{3} + c x^{2}\right )} \sqrt {b x^{3} + a}, x\right ) \]