\[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{a \,x^{2} \sqrt {b \,x^{3}+a x}}-\frac {21 b^{\frac {3}{2}} x \left (b \,x^{2}+a \right )}{5 a^{3} \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {b \,x^{3}+a x}}-\frac {7 \sqrt {b \,x^{3}+a x}}{5 a^{2} x^{3}}+\frac {21 b \sqrt {b \,x^{3}+a x}}{5 a^{3} x}+\frac {21 b^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {11}{4}} \sqrt {b \,x^{3}+a x}}-\frac {21 b^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {x}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {11}{4}} \sqrt {b \,x^{3}+a x}} \]
command
integrate(1/x^2/(b*x^3+a*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 \, {\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (21 \, b^{2} x^{4} + 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{3} + a x}}{5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x}}{b^{2} x^{8} + 2 \, a b x^{6} + a^{2} x^{4}}, x\right ) \]