Integrand size = 85, antiderivative size = 168 \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {4 \left (-a b^2 x+2 a b x^2+b^2 x^2-a x^3-2 b x^3+x^4\right )^{3/4}}{(-a+x) (-b+x)^2}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right ) \] Output:
4*(-a*b^2*x+2*a*b*x^2-a*x^3+b^2*x^2-2*b*x^3+x^4)^(3/4)/(-a+x)/(-b+x)^2+2*d ^(1/4)*arctan(d^(1/4)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4)/x) -2*d^(1/4)*arctanh(d^(1/4)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/ 4)/x)
Time = 41.75 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {4 \sqrt {\frac {b-x}{a-x}} x+2 \sqrt [4]{d} \sqrt [4]{\frac {x}{-a+x}} (-b+x) \arctan \left (\frac {\sqrt [4]{d} \sqrt {\frac {b-x}{a-x}}}{\left (\frac {x}{-a+x}\right )^{3/4}}\right )-2 \sqrt [4]{d} \sqrt [4]{\frac {x}{-a+x}} (-b+x) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {\frac {b-x}{a-x}}}{\left (\frac {x}{-a+x}\right )^{3/4}}\right )}{\sqrt {\frac {b-x}{a-x}} \sqrt [4]{(b-x)^2 x (-a+x)}} \] Input:
Integrate[(x^3*(-3*a*b + (a + 2*b)*x))/((-a + x)*(-b + x)*(x*(-a + x)*(-b + x)^2)^(1/4)*(-(a*b^2*d) + b*(2*a + b)*d*x - (a + 2*b)*d*x^2 + (-1 + d)*x ^3)),x]
Output:
(4*Sqrt[(b - x)/(a - x)]*x + 2*d^(1/4)*(x/(-a + x))^(1/4)*(-b + x)*ArcTan[ (d^(1/4)*Sqrt[(b - x)/(a - x)])/(x/(-a + x))^(3/4)] - 2*d^(1/4)*(x/(-a + x ))^(1/4)*(-b + x)*ArcTanh[(d^(1/4)*Sqrt[(b - x)/(a - x)])/(x/(-a + x))^(3/ 4)])/(Sqrt[(b - x)/(a - x)]*((b - x)^2*x*(-a + x))^(1/4))
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.
| \(\displaystyle \int \frac {x^3 (x (a+2 b)-3 a b)}{(x-a) (x-b) \sqrt [4]{x (x-a) (x-b)^2} \left (-a b^2 d-d x^2 (a+2 b)+b d x (2 a+b)+(d-1) x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{11/4} (3 a b-(a+2 b) x)}{(a-x) (b-x) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left ((1-d) x^3+(a+2 b) d x^2-b (2 a+b) d x+a b^2 d\right )}dx}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x) (b-x) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left ((1-d) x^3+(a+2 b) d x^2-b (2 a+b) d x+a b^2 d\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {\sqrt {x} a}{(a-x) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}+\frac {2 b \sqrt {x}}{(b-x) \sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2}}+\frac {d (b-x) \sqrt {x} ((a+2 b) x-3 a b)}{\sqrt [4]{x^3-(a+2 b) x^2+b (2 a+b) x-a b^2} \left ((1-d) x^3+a \left (\frac {2 b}{a}+1\right ) d x^2-2 a b \left (\frac {b}{2 a}+1\right ) d x+a b^2 d\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} ((a+2 b) x-3 a b)}{(a-x) (b-x) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int -\frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left (-\left ((1-d) x^3\right )-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left (-\left ((1-d) x^3\right )-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left (-\left ((1-d) x^3\right )-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {(a+2 b) x^{3/2}}{(1-d) (a-x)^{5/4} (b-x)^{3/2}}+\frac {\left (-a d \left (d a^2+b (d+3) a+4 b^2 d\right ) b^2+d \left (2 d a^3+b (2 d+7) a^2+b^2 (7 d+5) a+4 b^3 d\right ) x b-(a+2 b) d \left (d a^2+b (5-d) a+b^2 (3 d+1)\right ) x^2\right ) \sqrt {x}}{(d-1)^2 (a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}-\frac {\left (d a^2+b (d+3) a+4 b^2 d\right ) \sqrt {x}}{(1-d)^2 (a-x)^{5/4} (b-x)^{3/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a-x} \sqrt {b-x} \sqrt [4]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \frac {x^{7/2} (3 a b-(a+2 b) x)}{(a-x)^{5/4} (b-x)^{3/2} \left ((d-1) x^3-2 b d x^2+b^2 d x-a d (b-x)^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \sqrt [4]{-\left (x (a-x) (b-x)^2\right )}}\) |
Input:
Int[(x^3*(-3*a*b + (a + 2*b)*x))/((-a + x)*(-b + x)*(x*(-a + x)*(-b + x)^2 )^(1/4)*(-(a*b^2*d) + b*(2*a + b)*d*x - (a + 2*b)*d*x^2 + (-1 + d)*x^3)),x ]
Output:
$Aborted
Time = 2.13 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right ) \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}+\ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{-x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}\right ) \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}-4 x \left (\frac {1}{d}\right )^{\frac {1}{4}}}{\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}} \left (\frac {1}{d}\right )^{\frac {1}{4}}}\) | \(153\) |
Input:
int(x^3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-a*b^2 *d+b*(2*a+b)*d*x-(a+2*b)*d*x^2+(-1+d)*x^3),x,method=_RETURNVERBOSE)
Output:
-(-2*arctan((-x*(a-x)*(b-x)^2)^(1/4)/x/(1/d)^(1/4))*(-x*(a-x)*(b-x)^2)^(1/ 4)+ln((x*(1/d)^(1/4)+(-x*(a-x)*(b-x)^2)^(1/4))/(-x*(1/d)^(1/4)+(-x*(a-x)*( b-x)^2)^(1/4)))*(-x*(a-x)*(b-x)^2)^(1/4)-4*x*(1/d)^(1/4))/(-x*(a-x)*(b-x)^ 2)^(1/4)/(1/d)^(1/4)
Timed out. \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x^3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)^2)^(1/4)/( -a*b^2*d+b*(2*a+b)*d*x-(a+2*b)*d*x^2+(-1+d)*x^3),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x**3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)**2)**(1/4 )/(-a*b**2*d+b*(2*a+b)*d*x-(a+2*b)*d*x**2+(-1+d)*x**3),x)
Output:
Timed out
\[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\int { \frac {{\left (3 \, a b - {\left (a + 2 \, b\right )} x\right )} x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x + {\left (a + 2 \, b\right )} d x^{2} - {\left (d - 1\right )} x^{3}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \] Input:
integrate(x^3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)^2)^(1/4)/( -a*b^2*d+b*(2*a+b)*d*x-(a+2*b)*d*x^2+(-1+d)*x^3),x, algorithm="maxima")
Output:
integrate((3*a*b - (a + 2*b)*x)*x^3/((a*b^2*d - (2*a + b)*b*d*x + (a + 2*b )*d*x^2 - (d - 1)*x^3)*(-(a - x)*(b - x)^2*x)^(1/4)*(a - x)*(b - x)), x)
\[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\int { \frac {{\left (3 \, a b - {\left (a + 2 \, b\right )} x\right )} x^{3}}{{\left (a b^{2} d - {\left (2 \, a + b\right )} b d x + {\left (a + 2 \, b\right )} d x^{2} - {\left (d - 1\right )} x^{3}\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{4}} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \] Input:
integrate(x^3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)^2)^(1/4)/( -a*b^2*d+b*(2*a+b)*d*x-(a+2*b)*d*x^2+(-1+d)*x^3),x, algorithm="giac")
Output:
integrate((3*a*b - (a + 2*b)*x)*x^3/((a*b^2*d - (2*a + b)*b*d*x + (a + 2*b )*d*x^2 - (d - 1)*x^3)*(-(a - x)*(b - x)^2*x)^(1/4)*(a - x)*(b - x)), x)
Timed out. \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=-\int \frac {x^3\,\left (3\,a\,b-x\,\left (a+2\,b\right )\right )}{\left (a-x\right )\,\left (b-x\right )\,{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (x^3\,\left (d-1\right )-d\,x^2\,\left (a+2\,b\right )-a\,b^2\,d+b\,d\,x\,\left (2\,a+b\right )\right )} \,d x \] Input:
int(-(x^3*(3*a*b - x*(a + 2*b)))/((a - x)*(b - x)*(-x*(a - x)*(b - x)^2)^( 1/4)*(x^3*(d - 1) - d*x^2*(a + 2*b) - a*b^2*d + b*d*x*(2*a + b))),x)
Output:
-int((x^3*(3*a*b - x*(a + 2*b)))/((a - x)*(b - x)*(-x*(a - x)*(b - x)^2)^( 1/4)*(x^3*(d - 1) - d*x^2*(a + 2*b) - a*b^2*d + b*d*x*(2*a + b))), x)
\[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\text {too large to display} \] Input:
int(x^3*(-3*a*b+(a+2*b)*x)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-a*b^2 *d+b*(2*a+b)*d*x-(a+2*b)*d*x^2+(-1+d)*x^3),x)
Output:
- int(x**4/(x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a**2*b**3*d - 3*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2 *x - 2*b*x**2 + x**3)**(1/4)*a**2*b**2*d*x + 3*x**(1/4)*( - a*b**2 + 2*a*b *x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a**2*b*d*x**2 - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a**2*d*x**3 - 2*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**( 1/4)*a*b**3*d*x + 6*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b* x**2 + x**3)**(1/4)*a*b**2*d*x**2 - 6*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x* *2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a*b*d*x**3 + x**(1/4)*( - a*b**2 + 2 *a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a*b*x**3 + 2*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a*d*x**4 - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4) *a*x**4 + x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x** 3)**(1/4)*b**3*d*x**2 - 3*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b**2*d*x**3 + 3*x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b*d*x**4 - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*b*x**4 - x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*d*x**5 + x** (1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*x** 5),x)*a - 2*int(x**4/(x**(1/4)*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x -...