\[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ \frac {d \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{2 b e}-\frac {\left (-3 a d +4 b c \right ) \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right ) \sqrt {e}}{4 b^{\frac {7}{4}}}+\frac {\left (-3 a d +4 b c \right ) \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right ) \sqrt {e}}{4 b^{\frac {7}{4}}} \]
command
integrate((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(3/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{8} \, {\left (4 \, c {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )} - d {\left (\frac {3 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )}}{b} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} a}{{\left (b^{2} - \frac {{\left (b x^{2} + a\right )} b}{x^{2}}\right )} \sqrt {x}}\right )}\right )} e^{\frac {1}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (d x^{2} + c\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]________________________________________________________________________________________