\[ \int x^3 \sqrt {a x+b x^3} \, dx \]
Optimal antiderivative \[ -\frac {20 a^{2} \sqrt {b \,x^{3}+a x}}{231 b^{2}}+\frac {4 a \,x^{2} \sqrt {b \,x^{3}+a x}}{77 b}+\frac {2 x^{4} \sqrt {b \,x^{3}+a x}}{11}+\frac {10 a^{\frac {11}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{3}+a x}} \]
command
integrate(x^3*(b*x^3+a*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (10 \, a^{3} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, b^{3} x^{4} + 6 \, a b^{2} x^{2} - 10 \, a^{2} b\right )} \sqrt {b x^{3} + a x}\right )}}{231 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b x^{3} + a x} x^{3}, x\right ) \]