Integrand size = 108, antiderivative size = 60 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{(a-x)^2}\right )}{\sqrt {d}} \]
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 24.88 (sec) , antiderivative size = 3908, normalized size of antiderivative = 65.13 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Result too large to show} \]
Integrate[(a*(a*b + a*c - 3*b*c) + (-2*a^2 + a*b + a*c + 3*b*c)*x + (a - 2 *b - 2*c)*x^2 + x^3)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(-a^3 - b*c*d + (3* a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)),x]
((2*I)*Sqrt[(a - x)/(b - x)]*(b - x)*Sqrt[(-c + x)/(a - c)]*(3*(a - b)*(a - c)*d*(EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - El lipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d) *#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]) + (2*a - b - c)*d^2*(EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], ( a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]) - 2*(-2*a + b + c)^2*d*(EllipticPi[(a - b)/Root [a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^ 2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)]) - 3* (a - b)*(a - c)*d*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[(-a + x)/( a - b)]], (a - b)/(a - c)] - EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[ (-a + x)/(a - b)]], (a - b)/(a - c)]) - 3*(a - b)*(a - c)*(EllipticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)*#1 + d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + Ellip ticPi[(a - b)/Root[a^2*d - a*b*d - a*c*d + b*c*d + (-2*a*d + b*d + c*d)...
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.
| \(\displaystyle \int \frac {x \left (-2 a^2+a b+a c+3 b c\right )+x^2 (a-2 b-2 c)+a (a b+a c-3 b c)+x^3}{\sqrt {(x-a) (x-b) (x-c)} \left (-a^3+x \left (3 a^2+b d+c d\right )-x^2 (3 a+d)-b c d+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {-x^3-(a-2 (b+c)) x^2+\left (2 a^2-(b+c) a-3 b c\right ) x+a (3 b c-a (b+c))}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2+(-2 a+2 b+2 c) x+a b+a c-3 b c\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-a^3+x^3-(3 a+d) x^2-b c d+\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
Int[(a*(a*b + a*c - 3*b*c) + (-2*a^2 + a*b + a*c + 3*b*c)*x + (a - 2*b - 2 *c)*x^2 + x^3)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)),x]
3.8.78.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ p])*z^(q*FracPart[p]))) Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a , m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !F reeQ[z, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.79 (sec) , antiderivative size = 510, normalized size of antiderivative = 8.50
| method | result | size |
| default | \(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+d b +c d \right ) \textit {\_Z} -a^{3}-b c d \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -2 \underline {\hspace {1.25 ex}}\alpha ^{2} c +\underline {\hspace {1.25 ex}}\alpha ^{2} d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}+\underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a c +3 \underline {\hspace {1.25 ex}}\alpha b c -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{3}+a^{2} b +a^{2} c -3 a b c +b c d \right ) \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +d b +c^{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +d b +c^{2}\right ) \left (b -c \right )}{a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-d b -c d \right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}\, \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}\right )\) | \(510\) |
| elliptic | \(\frac {2 \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-c +x}{b -c}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}-2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+d b +c d \right ) \textit {\_Z} -a^{3}-b c d \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha ^{2} d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a c -3 \underline {\hspace {1.25 ex}}\alpha b c +\underline {\hspace {1.25 ex}}\alpha b d +\underline {\hspace {1.25 ex}}\alpha c d -a^{3}-a^{2} b -a^{2} c +3 a b c -b c d \right ) \left (b -c \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +d b +c^{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha c -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a c +d b +c^{2}\right ) \left (b -c \right )}{a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-d b -c d \right ) \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}\, \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}\right )\) | \(516\) |
int((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3)/((-a+ x)*(-b+x)*(-c+x))^(1/2)/(-a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x,m ethod=_RETURNVERBOSE)
2*(b-c)*((-c+x)/(b-c))^(1/2)*((-a+x)/(c-a))^(1/2)*((-b+x)/(-b+c))^(1/2)/(- a*b*c+a*b*x+a*c*x-a*x^2+b*c*x-b*x^2-c*x^2+x^3)^(1/2)*EllipticF(((-c+x)/(b- c))^(1/2),((-b+c)/(c-a))^(1/2))+2*sum((4*_alpha^2*a-2*_alpha^2*b-2*_alpha^ 2*c+_alpha^2*d-5*_alpha*a^2+_alpha*a*b+_alpha*a*c+3*_alpha*b*c-_alpha*b*d- _alpha*c*d+a^3+a^2*b+a^2*c-3*a*b*c+b*c*d)/(-3*_alpha^2+6*_alpha*a+2*_alpha *d-3*a^2-b*d-c*d)*(b-c)*((-c+x)/(b-c))^(1/2)*((-a+x)/(c-a))^(1/2)*((-b+x)/ (-b+c))^(1/2)/(-a*b*c+a*b*x+a*c*x-a*x^2+b*c*x-b*x^2-c*x^2+x^3)^(1/2)*(_alp ha^2-3*_alpha*a+_alpha*c-_alpha*d+3*a^2-3*a*c+b*d+c^2)/(a^3-3*a^2*c+3*a*c^ 2-c^3)*EllipticPi(((-c+x)/(b-c))^(1/2),(_alpha^2-3*_alpha*a+_alpha*c-_alph a*d+3*a^2-3*a*c+b*d+c^2)*(b-c)/(a^3-3*a^2*c+3*a*c^2-c^3),((-b+c)/(c-a))^(1 /2)),_alpha=RootOf(_Z^3+(-3*a-d)*_Z^2+(3*a^2+b*d+c*d)*_Z-a^3-b*c*d))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (49) = 98\).
Time = 26.54 (sec) , antiderivative size = 638, normalized size of antiderivative = 10.63 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{6} - 6 \, a^{3} b c d + b^{2} c^{2} d^{2} - 6 \, {\left (a - d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} - 6 \, {\left (3 \, a + b + c\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + {\left (b + c\right )} d^{2} - 3 \, {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d\right )} x^{3} + {\left (15 \, a^{4} + {\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 4 \, {\left (a^{4} - a b c d - {\left (4 \, a - d\right )} x^{3} + x^{4} + {\left (6 \, a^{2} - {\left (a + b + c\right )} d\right )} x^{2} - {\left (4 \, a^{3} - {\left (a b + {\left (a + b\right )} c\right )} d\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {d} - 2 \, {\left (3 \, a^{5} + {\left (b^{2} c + b c^{2}\right )} d^{2} - 3 \, {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}{a^{6} + 2 \, a^{3} b c d + b^{2} c^{2} d^{2} - 2 \, {\left (3 \, a + d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} + 2 \, {\left (3 \, a + b + c\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + {\left (b + c\right )} d^{2} + {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d\right )} x^{3} + {\left (15 \, a^{4} + {\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{2} + a b\right )} c\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} + {\left (b^{2} c + b c^{2}\right )} d^{2} + {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {{\left (a^{3} - b c d + {\left (3 \, a - d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} - {\left (b + c\right )} d\right )} x\right )} \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} \sqrt {-d}}{2 \, {\left (a^{2} b c d - {\left (2 \, a + b + c\right )} d x^{3} + d x^{4} + {\left (a^{2} + 2 \, a b + {\left (2 \, a + b\right )} c\right )} d x^{2} - {\left (a^{2} b + {\left (a^{2} + 2 \, a b\right )} c\right )} d x\right )}}\right )}{d}\right ] \]
integrate((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3) /((-a+x)*(-b+x)*(-c+x))^(1/2)/(-a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^ 3),x, algorithm="fricas")
[1/2*log((a^6 - 6*a^3*b*c*d + b^2*c^2*d^2 - 6*(a - d)*x^5 + x^6 + (15*a^2 - 6*(3*a + b + c)*d + d^2)*x^4 - 2*(10*a^3 + (b + c)*d^2 - 3*(3*a^2 + 3*a* b + (3*a + b)*c)*d)*x^3 + (15*a^4 + (b^2 + 4*b*c + c^2)*d^2 - 6*(a^3 + 3*a ^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 4*(a^4 - a*b*c*d - (4*a - d)*x^3 + x^4 + (6*a^2 - (a + b + c)*d)*x^2 - (4*a^3 - (a*b + (a + b)*c)*d)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sqrt(d) - 2*(3*a^5 + (b^2* c + b*c^2)*d^2 - 3*(a^3*b + (a^3 + 3*a^2*b)*c)*d)*x)/(a^6 + 2*a^3*b*c*d + b^2*c^2*d^2 - 2*(3*a + d)*x^5 + x^6 + (15*a^2 + 2*(3*a + b + c)*d + d^2)*x ^4 - 2*(10*a^3 + (b + c)*d^2 + (3*a^2 + 3*a*b + (3*a + b)*c)*d)*x^3 + (15* a^4 + (b^2 + 4*b*c + c^2)*d^2 + 2*(a^3 + 3*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 2*(3*a^5 + (b^2*c + b*c^2)*d^2 + (a^3*b + (a^3 + 3*a^2*b)*c)*d)*x))/sqr t(d), sqrt(-d)*arctan(-1/2*(a^3 - b*c*d + (3*a - d)*x^2 - x^3 - (3*a^2 - ( b + c)*d)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sq rt(-d)/(a^2*b*c*d - (2*a + b + c)*d*x^3 + d*x^4 + (a^2 + 2*a*b + (2*a + b) *c)*d*x^2 - (a^2*b + (a^2 + 2*a*b)*c)*d*x))/d]
Timed out. \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((a*(a*b+a*c-3*b*c)+(-2*a**2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x**2+x* *3)/((-a+x)*(-b+x)*(-c+x))**(1/2)/(-a**3-b*c*d+(3*a**2+b*d+c*d)*x-(3*a+d)* x**2+x**3),x)
\[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { -\frac {{\left (a - 2 \, b - 2 \, c\right )} x^{2} + x^{3} + {\left (a b + a c - 3 \, b c\right )} a - {\left (2 \, a^{2} - a b - a c - 3 \, b c\right )} x}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}}} \,d x } \]
integrate((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3) /((-a+x)*(-b+x)*(-c+x))^(1/2)/(-a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^ 3),x, algorithm="maxima")
-integrate(((a - 2*b - 2*c)*x^2 + x^3 + (a*b + a*c - 3*b*c)*a - (2*a^2 - a *b - a*c - 3*b*c)*x)/((a^3 + b*c*d + (3*a + d)*x^2 - x^3 - (3*a^2 + b*d + c*d)*x)*sqrt(-(a - x)*(b - x)*(c - x))), x)
Timed out. \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
integrate((a*(a*b+a*c-3*b*c)+(-2*a^2+a*b+a*c+3*b*c)*x+(a-2*b-2*c)*x^2+x^3) /((-a+x)*(-b+x)*(-c+x))^(1/2)/(-a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^ 3),x, algorithm="giac")
Time = 6.83 (sec) , antiderivative size = 946, normalized size of antiderivative = 15.77 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\left (\sum _{k=1}^3\left (-\frac {2\,\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (\frac {a-c}{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (a^2\,b+a^2\,c+4\,a\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-5\,a^2\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-2\,b\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-2\,c\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2+d\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2+a^3-3\,a\,b\,c+b\,c\,d+a\,b\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+a\,c\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+3\,b\,c\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-b\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )\right )}{\left (\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )-c\right )\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}\,\left (3\,a^2-6\,a\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+3\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+c\,d+3\,a^2\right )-b\,c\,d-a^3,z,k\right )+b\,d+c\,d\right )}\right )\right )+\frac {2\,\left (a-c\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}}{\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}} \]
int(-(a*(a*b + a*c - 3*b*c) + x*(a*b + a*c + 3*b*c - 2*a^2) + x^3 - x^2*(2 *b - a + 2*c))/((-(a - x)*(b - x)*(c - x))^(1/2)*(x^2*(3*a + d) - x*(b*d + c*d + 3*a^2) + a^3 - x^3 + b*c*d)),x)
symsum(-(2*(a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*ellipticPi((a - c)/(root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - c), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*(a^2*b + a^2*c + 4*a*root(z^3 - z^2*(3*a + d) + z*(b*d + c* d + 3*a^2) - b*c*d - a^3, z, k)^2 - 5*a^2*root(z^3 - z^2*(3*a + d) + z*(b* d + c*d + 3*a^2) - b*c*d - a^3, z, k) - 2*b*root(z^3 - z^2*(3*a + d) + z*( b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2 - 2*c*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2 + d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2 + a^3 - 3*a*b*c + b*c*d + a*b*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) + a*c*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k ) + 3*b*c*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - b*d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3 , z, k) - c*d*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a ^3, z, k)))/((root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a ^3, z, k) - c)*(-(a - x)*(b - x)*(c - x))^(1/2)*(b*d + c*d - 6*a*root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) - 2*d*root(z^ 3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k) + 3*a^2 + 3 *root(z^3 - z^2*(3*a + d) + z*(b*d + c*d + 3*a^2) - b*c*d - a^3, z, k)^2)) , k, 1, 3) + (2*(a - c)*ellipticF(asin((-(c - x)/(a - c))^(1/2)), (a - ...