\[ \int x^{3/2} \left (b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {5}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{17}+\frac {56 b^{4} x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{1105 c^{\frac {5}{2}} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {8 b^{2} x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{663 c}+\frac {12 b \,x^{\frac {9}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{221}-\frac {56 b^{3} \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}{3315 c^{2}}-\frac {56 b^{\frac {17}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{1105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {28 b^{\frac {17}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{1105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(3/2)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (84 \, b^{4} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (195 \, c^{4} x^{6} + 285 \, b c^{3} x^{4} + 20 \, b^{2} c^{2} x^{2} - 28 \, b^{3} c\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{3315 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c x^{5} + b x^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]