\[ \int x^3 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x^{4} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{11}+\frac {6 a \sqrt {c \,x^{2}}\, \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{55 b \,c^{2}}-\frac {4 \,3^{\frac {3}{4}} a^{2} \EllipticF \left (\frac {a^{\frac {1}{3}} \left (1-\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}, i \sqrt {3}+2 i\right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+b^{\frac {2}{3}} c \,x^{2}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}}{55 b^{\frac {4}{3}} c^{2} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}} \]
command
integrate(x^3*(a+b*(c*x^2)^(3/2))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (6 \, \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} a^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right ) - {\left (5 \, b^{2} c^{3} x^{4} + 3 \, \sqrt {c x^{2}} a b c\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right )}}{55 \, b^{2} c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {\sqrt {c x^{2}} b c x^{2} + a} x^{3}, x\right ) \]