3.20.36 \(\int \frac {(a^2-2 a x+x^2) (-a b-a c+3 b c+2 (a-b-c) x+x^2)}{((-a+x) (-b+x) (-c+x))^{3/4} (-b c-a^3 d+(b+c+3 a^2 d) x-(1+3 a d) x^2+d x^3)} \, dx\) [1936]

3.20.36.1 Optimal result

Integrand size = 99, antiderivative size = 135 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}} \]

output

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

3.20.36.2 Mathematica [F]

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

input

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

output

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

3.20.36.3 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.

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

\(\Big \downarrow \) 7269

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2004

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \frac {(x-a)^{5/4} \left (x^2+2 (a-b-c) x+3 b c-a (b+c)\right )}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{3/4} (x-b)^{3/4} (x-c)^{3/4} \int \left (\frac {(x-a)^{5/4} x^2}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {2 (a-b-c) (x-a)^{5/4} x}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}+\frac {a (-b-c) \left (1-\frac {3 b c}{a b+a c}\right ) (x-a)^{5/4}}{(x-b)^{3/4} (x-c)^{3/4} \left (d a^3-d x^3+(3 a d+1) x^2+b c-\left (3 d a^2+b+c\right ) x\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{3/4}}\)

input

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

output

$Aborted
 

3.20.36.3.1 Defintions of rubi rules used

rule 25

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

rule 2004

Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) 
, x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b 
, c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
 

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] 
]
 

3.20.36.4 Maple [F]

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

input

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

output

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

3.20.36.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.20.36.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.20.36.7 Maxima [F]

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

input

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

output

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

3.20.36.8 Giac [F]

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

input

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

output

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

3.20.36.9 Mupad [F(-1)]

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

input

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

output

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