\[ \int \frac {1}{\sqrt {a x+b x^4}} \, dx \]
Optimal antiderivative \[ \frac {x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right ) \EllipticF \left (\sqrt {1-\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}, \frac {\sqrt {6}}{4}+\frac {\sqrt {2}}{4}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{3 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {1}{3}} \sqrt {b \,x^{4}+a x}\, \sqrt {\frac {b^{\frac {1}{3}} x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(1/(b*x^4+a*x)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{3} \, \sqrt {b x^{4} + a x} x - \frac {a \arctan \left (\frac {\sqrt {b + \frac {a}{x^{3}}}}{\sqrt {-b}}\right )}{3 \, \sqrt {-b}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {b x^{4} + a x}}\,{d x} \]________________________________________________________________________________________