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