\[ \int x \sqrt {a x+b x^3} \, dx \]
Optimal antiderivative \[ \frac {4 a \sqrt {b \,x^{3}+a x}}{21 b}+\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7}-\frac {2 a^{\frac {7}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {5}{4}} \sqrt {b \,x^{3}+a x}} \]
command
integrate(x*(b*x^3+a*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, a^{2} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (3 \, b^{2} x^{2} + 2 \, a b\right )} \sqrt {b x^{3} + a x}\right )}}{21 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b x^{3} + a x} x, x\right ) \]