Integrand size = 108, antiderivative size = 60 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 24.01 (sec) , antiderivative size = 4752, normalized size of antiderivative = 79.20 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Result too large to show} \]
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)]*(-(b*c) - a^3*d + ( b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]
((2*I)*(-a + x)*Sqrt[(-c + x)/(a - c)]*(-2*a*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + b*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + c*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - 3*a^2*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*a*b*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 3*a*c*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b )/(a - c)] - 3*b*c*d*EllipticF[I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/ (a - c)] + 2*a*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/ (a - c)] - b*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c )*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] - c*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)* #1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 11*a^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b) /(a - c)] - 11*a*b*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + (-2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]], (a - b)/(a - c)] + 2*b^2*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c + b*c + ( -2*a + b + c)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[(-a + x)/(a - b)]] , (a - b)/(a - c)] - 11*a*c*d*EllipticPi[(a - b)/Root[a^2 - a*b - a*c +...
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.
| \(\displaystyle \int \frac {x \left (-2 a^2+a b+a c+3 b c\right )+x^2 (a-2 b-2 c)+a (a b+a c-3 b c)+x^3}{\sqrt {(x-a) (x-b) (x-c)} \left (a^3 (-d)+x \left (3 a^2 d+b+c\right )-x^2 (3 a d+1)-b c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {-x^3-(a-2 (b+c)) x^2+\left (2 a^2-(b+c) a-3 b c\right ) x+a (3 b c-a (b+c))}{\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2+(-2 a+2 b+2 c) x+a b+a c-3 b c\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \frac {\sqrt {x-a} \left (-x^2-2 (a-b-c) x-3 b c+a (b+c)\right )}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {x-a} \sqrt {x-b} \sqrt {x-c} \int \left (\frac {\sqrt {x-a} x^2}{\sqrt {x-b} \sqrt {x-c} \left (-d a^3+d x^3-(3 a d+1) x^2-b c+\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (-a+b+c) \sqrt {x-a} x}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) \sqrt {x-a}}{\sqrt {x-b} \sqrt {x-c} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{\sqrt {-((a-x) (b-x) (c-x))}}\) |
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)]*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]
3.8.81.3.1 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 = 3.95 (sec) , antiderivative size = 534, normalized size of antiderivative = 8.90
| 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 )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -d \,a^{3}-b c \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha a c d +3 \underline {\hspace {1.25 ex}}\alpha b c d +d \,a^{3}+a^{2} b d +a^{2} c d -3 a b c d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 c a d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 c a d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b -c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\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}}}\right )}{d^{2}}\) | \(534\) |
| 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 )}{d \sqrt {-a b c +a b x +a c x -a \,x^{2}+b c x -b \,x^{2}-c \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b +c \right ) \textit {\_Z} -d \,a^{3}-b c \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha a c d -3 \underline {\hspace {1.25 ex}}\alpha b c d -d \,a^{3}-a^{2} b d -a^{2} c d +3 a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \left (b -c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 c a d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {\frac {-c +x}{b -c}}\, \sqrt {\frac {-a +x}{c -a}}\, \sqrt {\frac {-b +x}{-b +c}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {-c +x}{b -c}}, \frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha c d +3 a^{2} d -3 c a d +c^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \left (b -c \right )}{d \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )}, \sqrt {\frac {-b +c}{c -a}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b +c \right ) \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\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}}}\right )}{d^{2}}\) | \(536\) |
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)/(-b*c-d*a^3+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d*x^3),x ,method=_RETURNVERBOSE)
2/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)*EllipticF(((-c+x)/( b-c))^(1/2),((-b+c)/(c-a))^(1/2))+2/d^2*sum((4*_alpha^2*a*d-2*_alpha^2*b*d -2*_alpha^2*c*d-5*_alpha*a^2*d+_alpha*a*b*d+_alpha*a*c*d+3*_alpha*b*c*d+a^ 3*d+a^2*b*d+a^2*c*d-3*a*b*c*d+_alpha^2-_alpha*b-_alpha*c+b*c)/(-3*_alpha^2 *d+6*_alpha*a*d-3*a^2*d+2*_alpha-b-c)*(b-c)*(_alpha^2*d-3*_alpha*a*d+_alph a*c*d+3*a^2*d-3*a*c*d+c^2*d-_alpha+b)/(a^3-3*a^2*c+3*a*c^2-c^3)*((-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)*EllipticPi(((-c+x)/(b-c))^(1/2),(_alpha ^2*d-3*_alpha*a*d+_alpha*c*d+3*a^2*d-3*a*c*d+c^2*d-_alpha+b)*(b-c)/d/(a^3- 3*a^2*c+3*a*c^2-c^3),((-b+c)/(c-a))^(1/2)),_alpha=RootOf(d*_Z^3+(-3*a*d-1) *_Z^2+(3*a^2*d+b+c)*_Z-d*a^3-b*c))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (49) = 98\).
Time = 49.38 (sec) , antiderivative size = 651, normalized size of antiderivative = 10.85 \[ \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 (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, a^{3} b c d - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} - 3 \, {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{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} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} - a b c + {\left (6 \, a^{2} d - a - b - c\right )} x^{2} - {\left (4 \, a^{3} d - a b - {\left (a + b\right )} c\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} d^{2} + b^{2} c + b c^{2} - 3 \, {\left (a^{3} b + {\left (a^{3} + 3 \, a^{2} b\right )} c\right )} d\right )} x}{a^{6} d^{2} + d^{2} x^{6} + 2 \, a^{3} b c d - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b + c\right )} d + 1\right )} x^{4} + b^{2} c^{2} - 2 \, {\left (10 \, a^{3} d^{2} + {\left (3 \, a^{2} + 3 \, a b + {\left (3 \, a + b\right )} c\right )} d + b + c\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 4 \, b c + c^{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} d^{2} + b^{2} c + b c^{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} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - b c - {\left (3 \, a^{2} d - b - c\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)/(-b*c-a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d* x^3),x, algorithm="fricas")
[1/2*log((a^6*d^2 + d^2*x^6 - 6*a^3*b*c*d - 6*(a*d^2 - d)*x^5 + (15*a^2*d^ 2 - 6*(3*a + b + c)*d + 1)*x^4 + b^2*c^2 - 2*(10*a^3*d^2 - 3*(3*a^2 + 3*a* b + (3*a + b)*c)*d + b + c)*x^3 + (15*a^4*d^2 + b^2 + 4*b*c + c^2 - 6*(a^3 + 3*a^2*b + 3*(a^2 + a*b)*c)*d)*x^2 - 4*(a^4*d + d*x^4 - (4*a*d - 1)*x^3 - a*b*c + (6*a^2*d - a - b - c)*x^2 - (4*a^3*d - a*b - (a + b)*c)*x)*sqrt( -a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b)*c)*x)*sqrt(d) - 2*(3*a^5*d ^2 + b^2*c + b*c^2 - 3*(a^3*b + (a^3 + 3*a^2*b)*c)*d)*x)/(a^6*d^2 + d^2*x^ 6 + 2*a^3*b*c*d - 2*(3*a*d^2 + d)*x^5 + (15*a^2*d^2 + 2*(3*a + b + c)*d + 1)*x^4 + b^2*c^2 - 2*(10*a^3*d^2 + (3*a^2 + 3*a*b + (3*a + b)*c)*d + b + c )*x^3 + (15*a^4*d^2 + b^2 + 4*b*c + c^2 + 2*(a^3 + 3*a^2*b + 3*(a^2 + a*b) *c)*d)*x^2 - 2*(3*a^5*d^2 + b^2*c + b*c^2 + (a^3*b + (a^3 + 3*a^2*b)*c)*d) *x))/sqrt(d), sqrt(-d)*arctan(-1/2*(a^3*d - d*x^3 + (3*a*d - 1)*x^2 - b*c - (3*a^2*d - b - c)*x)*sqrt(-a*b*c - (a + b + c)*x^2 + x^3 + (a*b + (a + b )*c)*x)*sqrt(-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. \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
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)/(-b*c-a**3*d+(3*a**2*d+b+c)*x-(3*a*d+1)* x**2+d*x**3),x)
\[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { -\frac {{\left (a - 2 \, b - 2 \, c\right )} x^{2} + x^{3} + {\left (a b + a c - 3 \, b c\right )} a - {\left (2 \, a^{2} - a b - a c - 3 \, b c\right )} x}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}}} \,d x } \]
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)/(-b*c-a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d* 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*d - d*x^3 + (3*a*d + 1)*x^2 + b*c - (3*a^2*d + b + c)*x)*sqrt(-(a - x)*(b - x)*(c - x))), x)
Exception generated. \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Exception raised: TypeError} \]
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)/(-b*c-a^3*d+(3*a^2*d+b+c)*x-(3*a*d+1)*x^2+d* x^3),x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Time = 27.82 (sec) , antiderivative size = 569, normalized size of antiderivative = 9.48 \[ \int \frac {a (a b+a c-3 b c)+\left (-2 a^2+a b+a c+3 b c\right ) x+(a-2 b-2 c) x^2+x^3}{\sqrt {(-a+x) (-b+x) (-c+x)} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\frac {\ln \left (\frac {\left (a-b-c+x+a^2\,d+d\,x^2-2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,a\,d\,x\right )\,\left (b\,c^2+b^2\,c+a^4\,d-a\,x^2+2\,b\,x^2-b^2\,x+2\,c\,x^2-c^2\,x+2\,d\,x^4-x^3+a^5\,d^2-d^2\,x^5+3\,a^2\,d\,x^2+5\,a\,d^2\,x^4-5\,a^4\,d^2\,x-2\,a^2\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-a\,b\,c+a\,b\,x+a\,c\,x-3\,b\,c\,x-10\,a^2\,d^2\,x^3+10\,a^3\,d^2\,x^2-a^3\,b\,d-a^3\,c\,d-4\,a\,d\,x^3-2\,a^3\,d\,x-2\,b\,d\,x^3-2\,c\,d\,x^3+3\,a^2\,b\,c\,d+3\,a\,b\,d\,x^2+3\,a\,c\,d\,x^2+3\,b\,c\,d\,x^2+2\,a\,b\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}+2\,a\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-2\,b\,c\,\sqrt {d}\,\sqrt {-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}-6\,a\,b\,c\,d\,x\right )}{\left (b\,c-b\,x-c\,x+a^3\,d-d\,x^3+x^2+3\,a\,d\,x^2-3\,a^2\,d\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d-2\,a^2\,c\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2+4\,a\,b\,c\,d-2\,a\,b-2\,a\,c-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2-4\,b\,c\,d\,x+2\,b\,c+2\,b\,d\,x^2-2\,b\,x+c^2+2\,c\,d\,x^2-2\,c\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \]
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)*(b*c - x*(b + c + 3*a^2*d ) + a^3*d - d*x^3 + x^2*(3*a*d + 1))),x)
log(((a - b - c + x + a^2*d + d*x^2 - 2*d^(1/2)*(-(a - x)*(b - x)*(c - x)) ^(1/2) - 2*a*d*x)*(b*c^2 + b^2*c + a^4*d - a*x^2 + 2*b*x^2 - b^2*x + 2*c*x ^2 - c^2*x + 2*d*x^4 - x^3 + a^5*d^2 - d^2*x^5 + 3*a^2*d*x^2 + 5*a*d^2*x^4 - 5*a^4*d^2*x - 2*a^2*d^(1/2)*(-(a - x)*(b - x)*(c - x))^(1/2) - a*b*c + a*b*x + a*c*x - 3*b*c*x - 10*a^2*d^2*x^3 + 10*a^3*d^2*x^2 - a^3*b*d - a^3* c*d - 4*a*d*x^3 - 2*a^3*d*x - 2*b*d*x^3 - 2*c*d*x^3 + 3*a^2*b*c*d + 3*a*b* d*x^2 + 3*a*c*d*x^2 + 3*b*c*d*x^2 + 2*a*b*d^(1/2)*(-(a - x)*(b - x)*(c - x ))^(1/2) + 2*a*c*d^(1/2)*(-(a - x)*(b - x)*(c - x))^(1/2) - 2*b*c*d^(1/2)* (-(a - x)*(b - x)*(c - x))^(1/2) - 6*a*b*c*d*x))/((b*c - b*x - c*x + a^3*d - d*x^3 + x^2 + 3*a*d*x^2 - 3*a^2*d*x)*(2*b*c - 2*a*c - 2*a*b + 2*a*x - 2 *b*x - 2*c*x + 2*a^3*d - 2*d*x^3 + a^2 + b^2 + c^2 + x^2 + a^4*d^2 + d^2*x ^4 - 4*a*d^2*x^3 - 4*a^3*d^2*x + 6*a^2*d^2*x^2 - 2*a^2*b*d - 2*a^2*c*d + 2 *a*d*x^2 - 2*a^2*d*x + 2*b*d*x^2 + 2*c*d*x^2 - 4*b*c*d*x + 4*a*b*c*d)))/d^ (1/2)