\[ \int x^4 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x^{5} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{13}+\frac {6 a c \,x^{7} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{91 b \left (c \,x^{2}\right )^{\frac {5}{2}}}-\frac {24 a^{2} x^{5} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{91 b^{\frac {5}{3}} \left (c \,x^{2}\right )^{\frac {5}{2}} \left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )}-\frac {8 \,3^{\frac {3}{4}} a^{\frac {7}{3}} x^{5} \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 ) \sqrt {2}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{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}}}}{91 b^{\frac {5}{3}} \left (c \,x^{2}\right )^{\frac {5}{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}}}}+\frac {12 \,3^{\frac {1}{4}} a^{\frac {7}{3}} x^{5} \EllipticE \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}}}}{91 b^{\frac {5}{3}} \left (c \,x^{2}\right )^{\frac {5}{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^4*(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
\[ \frac {2 \, {\left (12 \, \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} a^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right )\right ) + {\left (7 \, b^{2} c^{3} x^{5} + 3 \, \sqrt {c x^{2}} a b c x\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right )}}{91 \, b^{2} c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {\sqrt {c x^{2}} b c x^{2} + a} x^{4}, x\right ) \]