3.18.87 \(\int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} (-a^3-b c d+(3 a^2+b d+c d) x-(3 a+d) x^2+x^3)} \, dx\) [1787]

3.18.87.1 Optimal result

Integrand size = 88, antiderivative size = 121 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]

output

-2*arctan(d^(1/4)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/4)/(a-x))/d 
^(3/4)+2*arctanh(d^(1/4)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/4)/( 
a-x))/d^(3/4)
 

3.18.87.2 Mathematica [A] (verified)

Time = 10.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{-a+x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{a-x}\right )\right )}{d^{3/4}} \]

input

Integrate[(-(a*b) - a*c + 3*b*c + 2*(a - b - c)*x + x^2)/(((-a + x)*(-b + 
x)*(-c + x))^(1/4)*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + 
 x^3)),x]
 

output

(2*(ArcTan[(d^(1/4)*((-a + x)*(-b + x)*(-c + x))^(1/4))/(-a + x)] + ArcTan 
h[(d^(1/4)*((-a + x)*(-b + x)*(-c + x))^(1/4))/(a - x)]))/d^(3/4)
 

3.18.87.3 Rubi [F]

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.

\(\displaystyle \int \frac {2 x (a-b-c)-a b-a c+3 b c+x^2}{\sqrt [4]{(x-a) (x-b) (x-c)} \left (-a^3+x \left (3 a^2+b d+c d\right )-x^2 (3 a+d)-b c d+x^3\right )} \, dx\)

\(\Big \downarrow \) 7269

\(\displaystyle \frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int -\frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \left (\frac {x^2}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {2 (a-b-c) x}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}\right )dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \int \frac {x^2+2 (a-b-c) x+3 b c-a (b+c)}{\sqrt [4]{x-a} \sqrt [4]{x-b} \sqrt [4]{x-c} \left (a^3-x^3+(3 a+d) x^2+b c d-\left (3 a^2+(b+c) d\right ) x\right )}dx}{\sqrt [4]{-((a-x) (b-x) (c-x))}}\)

input

Int[(-(a*b) - a*c + 3*b*c + 2*(a - b - c)*x + x^2)/(((-a + x)*(-b + x)*(-c 
 + x))^(1/4)*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)) 
,x]
 

output

$Aborted
 

3.18.87.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7269

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]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.18.87.4 Maple [F]

\[\int \frac {-a b -a c +3 b c +2 \left (a -b -c \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{4}} \left (-a^{3}-b c d +\left (3 a^{2}+b d +c d \right ) x -\left (3 a +d \right ) x^{2}+x^{3}\right )}d x\]

input

int((-a*b-a*c+3*b*c+2*(a-b-c)*x+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/4)/(-a^3-b* 
c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x)
 

output

int((-a*b-a*c+3*b*c+2*(a-b-c)*x+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/4)/(-a^3-b* 
c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x)
 

3.18.87.5 Fricas [F(-1)]

Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]

input

integrate((-a*b-a*c+3*b*c+2*(a-b-c)*x+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/4)/(- 
a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x, algorithm="fricas")
 

output

Timed out
 

3.18.87.6 Sympy [F(-1)]

Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]

input

integrate((-a*b-a*c+3*b*c+2*(a-b-c)*x+x**2)/((-a+x)*(-b+x)*(-c+x))**(1/4)/ 
(-a**3-b*c*d+(3*a**2+b*d+c*d)*x-(3*a+d)*x**2+x**3),x)
 

output

Timed out
 

3.18.87.7 Maxima [F]

\[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { \frac {a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{4}}} \,d x } \]

input

integrate((-a*b-a*c+3*b*c+2*(a-b-c)*x+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/4)/(- 
a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x, algorithm="maxima")
 

output

integrate((a*b + a*c - 3*b*c - 2*(a - b - c)*x - x^2)/((a^3 + b*c*d + (3*a 
 + d)*x^2 - x^3 - (3*a^2 + b*d + c*d)*x)*(-(a - x)*(b - x)*(c - x))^(1/4)) 
, x)
 

3.18.87.8 Giac [F]

\[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { \frac {a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{4}}} \,d x } \]

input

integrate((-a*b-a*c+3*b*c+2*(a-b-c)*x+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/4)/(- 
a^3-b*c*d+(3*a^2+b*d+c*d)*x-(3*a+d)*x^2+x^3),x, algorithm="giac")
 

output

integrate((a*b + a*c - 3*b*c - 2*(a - b - c)*x - x^2)/((a^3 + b*c*d + (3*a 
 + d)*x^2 - x^3 - (3*a^2 + b*d + c*d)*x)*(-(a - x)*(b - x)*(c - x))^(1/4)) 
, x)
 

3.18.87.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int \frac {a\,b+a\,c-3\,b\,c+2\,x\,\left (b-a+c\right )-x^2}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{1/4}\,\left (x^2\,\left (3\,a+d\right )-x\,\left (3\,a^2+b\,d+c\,d\right )+a^3-x^3+b\,c\,d\right )} \,d x \]

input

int((a*b + a*c - 3*b*c + 2*x*(b - a + c) - x^2)/((-(a - x)*(b - x)*(c - x) 
)^(1/4)*(x^2*(3*a + d) - x*(b*d + c*d + 3*a^2) + a^3 - x^3 + b*c*d)),x)
 

output

int((a*b + a*c - 3*b*c + 2*x*(b - a + c) - x^2)/((-(a - x)*(b - x)*(c - x) 
)^(1/4)*(x^2*(3*a + d) - x*(b*d + c*d + 3*a^2) + a^3 - x^3 + b*c*d)), x)