\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{3}+a x \right )^{\frac {3}{2}}}{9}+\frac {8 a^{2} x \left (b \,x^{2}+a \right )}{15 \sqrt {b}\, \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {b \,x^{3}+a x}}+\frac {4 a x \sqrt {b \,x^{3}+a x}}{15}-\frac {8 a^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {3}{4}} \sqrt {b \,x^{3}+a x}}+\frac {4 a^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {3}{4}} \sqrt {b \,x^{3}+a x}} \]
command
integrate((b*x^3+a*x)^(3/2)/x,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, a^{2} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, b^{2} x^{3} + 11 \, a b x\right )} \sqrt {b x^{3} + a x}\right )}}{45 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b x^{3} + a x} {\left (b x^{2} + a\right )}, x\right ) \]