\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {16 \arctanh \left (\frac {\sqrt {e}\, \sqrt {b x +a}}{\sqrt {b}\, \sqrt {e x +d}}\right )}{\sqrt {b}\, \sqrt {e}}+\frac {2 d^{2} \sqrt {b x +a}}{\left (-a e +b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 d \left (-2 a e +3 b d \right ) \sqrt {b x +a}}{\left (-a e +b d \right )^{2} \sqrt {e x +d}} \]
command
integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(5/2)/(b*x+a)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, \sqrt {b} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{{\left | b \right |}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (3 \, b^{6} d^{2} e^{2} - 2 \, a b^{5} d e^{3}\right )} {\left (b x + a\right )}}{b^{4} d^{2} {\left | b \right |} e - 2 \, a b^{3} d {\left | b \right |} e^{2} + a^{2} b^{2} {\left | b \right |} e^{3}} + \frac {7 \, b^{7} d^{3} e - 11 \, a b^{6} d^{2} e^{2} + 4 \, a^{2} b^{5} d e^{3}}{b^{4} d^{2} {\left | b \right |} e - 2 \, a b^{3} d {\left | b \right |} e^{2} + a^{2} b^{2} {\left | b \right |} e^{3}}\right )}}{{\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________