\[ \int \frac {1}{x \left (a x+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{a x \sqrt {b \,x^{3}+a x}}-\frac {5 \sqrt {b \,x^{3}+a x}}{3 a^{2} x^{2}}-\frac {5 b^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {9}{4}} \sqrt {b \,x^{3}+a x}} \]
command
integrate(1/x/(b*x^3+a*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (b x^{4} + a x^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + \sqrt {b x^{3} + a x} {\left (5 \, b x^{2} + 2 \, a\right )}}{3 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x}}{b^{2} x^{7} + 2 \, a b x^{5} + a^{2} x^{3}}, x\right ) \]