\[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {315 b^{4} \arctanh \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^{\frac {11}{2}}}-\frac {4 x^{\frac {5}{2}}}{a \sqrt {b \sqrt {x}+a x}}-\frac {315 b^{3} \sqrt {b \sqrt {x}+a x}}{32 a^{5}}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^{3}}+\frac {9 x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}{2 a^{2}}+\frac {105 b^{2} \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{16 a^{4}} \]
command
integrate(x^(5/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{32} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, \sqrt {x} {\left (\frac {2 \, \sqrt {x}}{a^{2}} - \frac {5 \, b}{a^{3}}\right )} + \frac {41 \, b^{2}}{a^{4}}\right )} \sqrt {x} - \frac {187 \, b^{3}}{a^{5}}\right )} - \frac {315 \, b^{4} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{64 \, a^{\frac {11}{2}}} - \frac {4 \, b^{5}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________