\[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate((a*x^2+b*x)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{32 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{32 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{64 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{64 \, \left (-a\right )^{\frac {3}{4}}} - \frac {{\left (8 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{2} b - 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{3} b - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{2} + 5 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{2}\right )} x^{2}}{8 \, b^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________