Integrand size = 122, antiderivative size = 113 \[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{a-x}\right )}{d^{3/4}} \]
-2*arctan(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(3/4)/(a-x))/d^( 3/4)+2*arctanh(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(3/4)/(a-x) )/d^(3/4)
Time = 4.45 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.37 \[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\frac {\sqrt {2} (a-x)^{3/4} (b-x)^{3/2} \left (\arctan \left (\frac {\sqrt {a-x}+\sqrt {d} (-b+x)^3}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} (b-x)^{3/2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} (b-x)^{3/2}}{\sqrt {a-x}+\sqrt {d} (b-x)^3}\right )\right )}{d^{3/4} \left ((b-x)^2 (-a+x)\right )^{3/4}} \]
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]
(Sqrt[2]*(a - x)^(3/4)*(b - x)^(3/2)*(ArcTan[(Sqrt[a - x] + Sqrt[d]*(-b + x)^3)/(Sqrt[2]*d^(1/4)*(a - x)^(1/4)*(b - x)^(3/2))] + ArcTanh[(Sqrt[2]*d^ (1/4)*(a - x)^(1/4)*(b - x)^(3/2))/(Sqrt[a - x] + Sqrt[d]*(b - x)^3)]))/(d ^(3/4)*((b - x)^2*(-a + x))^(3/4))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right ) (-6 a+b+5 x)}{\left ((x-a) (x-b)^2\right )^{3/4} \left (a+b^6 d-x \left (6 b^5 d+1\right )+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {(x-b)^5 (-6 a+b+5 x)}{\left ((x-a) (x-b)^2\right )^{3/4} \left (a+b^6 d-x \left (6 b^5 d+1\right )+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )}dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{3/4} (x-b)^{3/2} \int -\frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d b^6+15 d x^2 b^4-20 d x^3 b^3+15 d x^4 b^2-6 d x^5 b+d x^6+a-\left (6 d b^5+1\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d b^6+15 d x^2 b^4-20 d x^3 b^3+15 d x^4 b^2-6 d x^5 b+d x^6+a-\left (6 d b^5+1\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) b (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}+\frac {5 x (x-b)^{7/2}}{(x-a)^{3/4} \left (-d x^6+6 b d x^5-15 b^2 d x^4+20 b^3 d x^3-15 b^4 d x^2+\left (6 d b^5+1\right ) x-a \left (\frac {d b^6}{a}+1\right )\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/2} \int \frac {(6 a-b-5 x) (x-b)^{7/2}}{(x-a)^{3/4} \left (d x^6-6 b d x^5+15 b^2 d x^4-20 b^3 d x^3+15 b^4 d x^2-\left (6 d b^5+1\right ) x+a \left (\frac {d b^6}{a}+1\right )\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4}}\) |
Int[((-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]
3.17.89.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {\left (-6 a +b +5 x \right ) \left (-b^{5}+5 b^{4} x -10 b^{3} x^{2}+10 b^{2} x^{3}-5 b \,x^{4}+x^{5}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (a +b^{6} d -\left (6 b^{5} d +1\right ) x +15 b^{4} d \,x^{2}-20 b^{3} d \,x^{3}+15 b^{2} d \,x^{4}-6 b d \,x^{5}+d \,x^{6}\right )}d x\]
int((-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-(6*b^5*d+1)*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)
int((-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-(6*b^5*d+1)*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)
Timed out. \[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\text {Timed out} \]
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-(6*b^5*d+1)*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, algorithm="fricas")
\[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\int \frac {\left (- b + x\right )^{5} \left (- 6 a + b + 5 x\right )}{\left (\left (- a + x\right ) \left (- b + x\right )^{2}\right )^{\frac {3}{4}} \left (a + b^{6} d - 6 b^{5} d x + 15 b^{4} d x^{2} - 20 b^{3} d x^{3} + 15 b^{2} d x^{4} - 6 b d x^{5} + d x^{6} - x\right )}\, dx \]
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-(6*b**5*d+1)*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)
Integral((-b + x)**5*(-6*a + b + 5*x)/(((-a + x)*(-b + x)**2)**(3/4)*(a + b**6*d - 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)), x)
\[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\int { \frac {{\left (b^{5} - 5 \, b^{4} x + 10 \, b^{3} x^{2} - 10 \, b^{2} x^{3} + 5 \, b x^{4} - x^{5}\right )} {\left (6 \, a - b - 5 \, x\right )}}{{\left (b^{6} d + 15 \, b^{4} d x^{2} - 20 \, b^{3} d x^{3} + 15 \, b^{2} d x^{4} - 6 \, b d x^{5} + d x^{6} - {\left (6 \, b^{5} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}} \,d x } \]
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-(6*b^5*d+1)*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, algorithm="maxima")
integrate((b^5 - 5*b^4*x + 10*b^3*x^2 - 10*b^2*x^3 + 5*b*x^4 - x^5)*(6*a - b - 5*x)/((b^6*d + 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 - (6*b^5*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(3/4)), x)
\[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\int { \frac {{\left (b^{5} - 5 \, b^{4} x + 10 \, b^{3} x^{2} - 10 \, b^{2} x^{3} + 5 \, b x^{4} - x^{5}\right )} {\left (6 \, a - b - 5 \, x\right )}}{{\left (b^{6} d + 15 \, b^{4} d x^{2} - 20 \, b^{3} d x^{3} + 15 \, b^{2} d x^{4} - 6 \, b d x^{5} + d x^{6} - {\left (6 \, b^{5} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}} \,d x } \]
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-(6*b^5*d+1)*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, algorithm="giac")
integrate((b^5 - 5*b^4*x + 10*b^3*x^2 - 10*b^2*x^3 + 5*b*x^4 - x^5)*(6*a - b - 5*x)/((b^6*d + 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 - (6*b^5*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(3/4)), x)
Timed out. \[ \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx=\int -\frac {\left (b-6\,a+5\,x\right )\,\left (b^5-5\,b^4\,x+10\,b^3\,x^2-10\,b^2\,x^3+5\,b\,x^4-x^5\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a+b^6\,d+d\,x^6-x\,\left (6\,d\,b^5+1\right )+15\,b^2\,d\,x^4-20\,b^3\,d\,x^3+15\,b^4\,d\,x^2-6\,b\,d\,x^5\right )} \,d x \]
int(-((b - 6*a + 5*x)*(5*b*x^4 - 5*b^4*x + b^5 - x^5 - 10*b^2*x^3 + 10*b^3 *x^2))/((-(a - x)*(b - x)^2)^(3/4)*(a + b^6*d + d*x^6 - x*(6*b^5*d + 1) + 15*b^2*d*x^4 - 20*b^3*d*x^3 + 15*b^4*d*x^2 - 6*b*d*x^5)),x)