\[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctan \left (\frac {d^{\frac {1}{4}} \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {3}{4}}}{a -x}\right )}{d^{\frac {3}{4}}}+\frac {2 \arctanh \left (\frac {d^{\frac {1}{4}} \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {3}{4}}}{a -x}\right )}{d^{\frac {3}{4}}} \]
command
Integrate[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-b + x)^2)^(3/4)*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {2} (a-x)^{3/4} (b-x)^{3/2} \left (\text {ArcTan}\left (\frac {\sqrt {a-x}+\sqrt {d} (-b+x)^3}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} (b-x)^{3/2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} (b-x)^{3/2}}{\sqrt {a-x}+\sqrt {d} (b-x)^3}\right )\right )}{d^{3/4} \left ((b-x)^2 (-a+x)\right )^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx \]________________________________________________________________________________________