Integrand size = 89, antiderivative size = 59 \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{x^2}\right )}{\sqrt {d}} \]
Time = 10.74 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x) (-c+x)}}{x^2}\right )}{\sqrt {d}} \]
Integrate[(x*(3*a*b*c - 2*(a*b + a*c + b*c)*x + (a + b + c)*x^2))/(Sqrt[x* (-a + x)*(-b + x)*(-c + x)]*(-(a*b*c*d) + (a*b + a*c + b*c)*d*x - (a + b + c)*d*x^2 + (-1 + d)*x^3)),x]
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.
| \(\displaystyle \int \frac {x \left (x^2 (a+b+c)-2 x (a b+a c+b c)+3 a b c\right )}{\sqrt {x (x-a) (x-b) (x-c)} \left (-d x^2 (a+b+c)+d x (a b+a c+b c)-a b c d+(d-1) x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int -\frac {\sqrt {x} \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {\sqrt {x} \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {-((a-x) (x-b) (x-c))} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+(a+b+c) d x^2-(b c+a (b+c)) d x+a b c d\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {-a-b-c}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {-\left (\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2\right )+\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x-a b c (a+b+c) d}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+a \left (\frac {b+c}{a}+1\right ) d x^2-a b \left (\frac {(a+b) c}{a b}+1\right ) d x+a b c d\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (d a^2+2 (b+c) a+2 b c+b^2 d+c^2 d\right ) x^2-\left ((b+c) d a^2+\left (d b^2+3 c b+c^2 d\right ) a+b c (b+c) d\right ) x+a b c (a+b+c) d}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {c x^3+b \left (x^3-2 c x^2\right )+a \left (x^3-2 b x^2-2 c x^2+3 b c x\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left ((1-d) x^3+c d x^2+b d (x-c) x+a d (b-x) (c-x)\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
Int[(x*(3*a*b*c - 2*(a*b + a*c + b*c)*x + (a + b + c)*x^2))/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(-(a*b*c*d) + (a*b + a*c + b*c)*d*x - (a + b + c)*d* x^2 + (-1 + d)*x^3)),x]
3.8.57.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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]
Time = 6.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x^{2}}\right )}{\sqrt {d}}\) | \(34\) |
| elliptic | \(\text {Expression too large to display}\) | \(665\) |
| default | \(\text {Expression too large to display}\) | \(668\) |
int(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x^2)/(x*(-a+x)*(-b+x)*(-c+x))^(1/ 2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x-(a+b+c)*d*x^2+(d-1)*x^3),x,method=_RETURNVE RBOSE)
Timed out. \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x^2)/(x*(-a+x)*(-b+x)*(-c+x ))^(1/2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x-(a+b+c)*d*x^2+(-1+d)*x^3),x, algorith m="fricas")
Timed out. \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x**2)/(x*(-a+x)*(-b+x)*(-c+ x))**(1/2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x-(a+b+c)*d*x**2+(-1+d)*x**3),x)
\[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=\int { -\frac {{\left (3 \, a b c + {\left (a + b + c\right )} x^{2} - 2 \, {\left (a b + a c + b c\right )} x\right )} x}{{\left (a b c d + {\left (a + b + c\right )} d x^{2} - {\left (d - 1\right )} x^{3} - {\left (a b + a c + b c\right )} d x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x}} \,d x } \]
integrate(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x^2)/(x*(-a+x)*(-b+x)*(-c+x ))^(1/2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x-(a+b+c)*d*x^2+(-1+d)*x^3),x, algorith m="maxima")
-integrate((3*a*b*c + (a + b + c)*x^2 - 2*(a*b + a*c + b*c)*x)*x/((a*b*c*d + (a + b + c)*d*x^2 - (d - 1)*x^3 - (a*b + a*c + b*c)*d*x)*sqrt(-(a - x)* (b - x)*(c - x)*x)), x)
Time = 0.54 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=\frac {2 \, {\left | d \right |} \arctan \left (\frac {\sqrt {-\frac {a b c}{x^{3}} + \frac {a b}{x^{2}} + \frac {a c}{x^{2}} + \frac {b c}{x^{2}} - \frac {a}{x} - \frac {b}{x} - \frac {c}{x} + 1}}{\sqrt {-\frac {1}{d}}}\right )}{\sqrt {-d} d} \]
integrate(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x^2)/(x*(-a+x)*(-b+x)*(-c+x ))^(1/2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x-(a+b+c)*d*x^2+(-1+d)*x^3),x, algorith m="giac")
2*abs(d)*arctan(sqrt(-a*b*c/x^3 + a*b/x^2 + a*c/x^2 + b*c/x^2 - a/x - b/x - c/x + 1)/sqrt(-1/d))/(sqrt(-d)*d)
Timed out. \[ \int \frac {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx=\int \frac {x\,\left (x^2\,\left (a+b+c\right )-2\,x\,\left (a\,b+a\,c+b\,c\right )+3\,a\,b\,c\right )}{\left (\left (d-1\right )\,x^3-d\,\left (a+b+c\right )\,x^2+d\,\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c\,d\right )\,\sqrt {-x\,\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}} \,d x \]
int((x*(x^2*(a + b + c) - 2*x*(a*b + a*c + b*c) + 3*a*b*c))/((x^3*(d - 1) + d*x*(a*b + a*c + b*c) - d*x^2*(a + b + c) - a*b*c*d)*(-x*(a - x)*(b - x) *(c - x))^(1/2)),x)