Integrand size = 83, antiderivative size = 113 \[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right )}{d^{3/4}} \]
2*arctan(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(b-x)^2)/d^ (3/4)-2*arctanh(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(b-x )^2)/d^(3/4)
Time = 10.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.60 \[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} (-a+x)}{\left ((b-x)^2 (-a+x)\right )^{3/4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} (-a+x)}{\left ((b-x)^2 (-a+x)\right )^{3/4}}\right )\right )}{d^{3/4}} \]
Integrate[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6 )),x]
(2*(ArcTan[(d^(1/4)*(-a + x))/((b - x)^2*(-a + x))^(3/4)] - ArcTanh[(d^(1/ 4)*(-a + x))/((b - x)^2*(-a + x))^(3/4)]))/d^(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.
| \(\displaystyle \int \frac {(x-b) (-6 a+b+5 x)}{\sqrt [4]{(x-a) (x-b)^2} \left (a d+b^6-x \left (6 b^5+d\right )+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [4]{x-a} \sqrt {x-b} \int -\frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (b^6+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6+a d-\left (6 b^5+d\right ) x\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (b^6+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6+a d-\left (6 b^5+d\right ) x\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6-\left (6 b^5+d\right ) x\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \frac {(6 a-b-5 x) \sqrt {x-b}}{\sqrt [4]{x-a} \left (\left (\frac {a d}{b^6}+1\right ) b^6-6 \left (\frac {d}{6 b^5}+1\right ) x b^5+15 x^2 b^4-20 x^3 b^3+15 x^4 b^2-6 x^5 b+x^6\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt {x-b} \int \left (\frac {\left (1-\frac {6 a}{b}\right ) \sqrt {x-b} b}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}+\frac {5 x \sqrt {x-b}}{\sqrt [4]{x-a} \left (-\left (\left (\frac {a d}{b^6}+1\right ) b^6\right )+6 \left (\frac {d}{6 b^5}+1\right ) x b^5-15 x^2 b^4+20 x^3 b^3-15 x^4 b^2+6 x^5 b-x^6\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\) |
Int[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6)),x]
3.17.88.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (-b +x \right ) \left (-6 a +b +5 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{6}+a d -\left (6 b^{5}+d \right ) x +15 b^{4} x^{2}-20 b^{3} x^{3}+15 b^{2} x^{4}-6 b \,x^{5}+x^{6}\right )}d x\]
int((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^ 4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+x^6),x)
int((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^ 4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+x^6),x)
Timed out. \[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\text {Timed out} \]
integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x +15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+x^6),x, algorithm="fricas")
\[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\int \frac {\left (- b + x\right ) \left (- 6 a + b + 5 x\right )}{\sqrt [4]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (a d + b^{6} - 6 b^{5} x + 15 b^{4} x^{2} - 20 b^{3} x^{3} + 15 b^{2} x^{4} - 6 b x^{5} - d x + x^{6}\right )}\, dx \]
integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)**2)**(1/4)/(b**6+a*d-(6*b**5+ d)*x+15*b**4*x**2-20*b**3*x**3+15*b**2*x**4-6*b*x**5+x**6),x)
Integral((-b + x)*(-6*a + b + 5*x)/(((-a + x)*(-b + x)**2)**(1/4)*(a*d + b **6 - 6*b**5*x + 15*b**4*x**2 - 20*b**3*x**3 + 15*b**2*x**4 - 6*b*x**5 - d *x + x**6)), x)
\[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\int { \frac {{\left (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}} \,d x } \]
integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x +15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+x^6),x, algorithm="maxima")
integrate((6*a - b - 5*x)*(b - x)/((b^6 + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2 *x^4 - 6*b*x^5 + x^6 + a*d - (6*b^5 + d)*x)*(-(a - x)*(b - x)^2)^(1/4)), x )
\[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\int { \frac {{\left (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}} \,d x } \]
integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x +15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+x^6),x, algorithm="giac")
integrate((6*a - b - 5*x)*(b - x)/((b^6 + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2 *x^4 - 6*b*x^5 + x^6 + a*d - (6*b^5 + d)*x)*(-(a - x)*(b - x)^2)^(1/4)), x )
Timed out. \[ \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (b-6\,a+5\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-6\,b\,x^5-x\,\left (6\,b^5+d\right )+b^6+x^6+15\,b^2\,x^4-20\,b^3\,x^3+15\,b^4\,x^2\right )} \,d x \]
int(-((b - x)*(b - 6*a + 5*x))/((-(a - x)*(b - x)^2)^(1/4)*(a*d - 6*b*x^5 - x*(d + 6*b^5) + b^6 + x^6 + 15*b^2*x^4 - 20*b^3*x^3 + 15*b^4*x^2)),x)