Integrand size = 74, antiderivative size = 289 \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-2+2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-1+k x}\right )}{b^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (-\sqrt [6]{b}+\sqrt [6]{b} k x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}}{1-2 k x+k^2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{5/6}} \] Output:
1/2*3^(1/2)*arctan(3^(1/2)*b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(2-2*k*x+b^( 1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(5/6)-1/2*3^(1/2)*arctan(3^(1/2)*b^(1/ 6)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(-2+2*k*x+b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3) ))/b^(5/6)+arctanh(b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(k*x-1))/b^(5/6)+1/2 *arctanh((-b^(1/6)+b^(1/6)*k*x)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(1-2*k*x+k^2*x^ 2+b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(2/3)))/b^(5/6)
\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \] Input:
Integrate[((-1 + x)*x*(1 + (-2 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4)),x]
Output:
Integrate[((-1 + x)*x*(1 + (-2 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4)), 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.
| \(\displaystyle \int \frac {(x-1) x ((k-2) x+1)}{\sqrt [3]{(1-x) x (1-k x)} \left (x^4 \left (k^4-b\right )+x^3 \left (2 b-4 k^3\right )+x^2 \left (6 k^2-b\right )-4 k x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {(1-x) x^{2/3} (1-(2-k) x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (-\left (\left (b-k^4\right ) x^4\right )+2 \left (b-2 k^3\right ) x^3-\left (b-6 k^2\right ) x^2-4 k x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(1-x) x^{2/3} (1-(2-k) x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (-\left (\left (b-k^4\right ) x^4\right )+2 \left (b-2 k^3\right ) x^3-\left (b-6 k^2\right ) x^2-4 k x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(1-x) x^{4/3} (1-(2-k) x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (-\left (\left (b-k^4\right ) x^4\right )+2 \left (b-2 k^3\right ) x^3-\left (b-6 k^2\right ) x^2-4 k x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {(1-x)^{2/3} x^{4/3} (1-(2-k) x)}{\sqrt [3]{1-k x} \left (-\left (\left (b-k^4\right ) x^4\right )+2 \left (b-2 k^3\right ) x^3-\left (b-6 k^2\right ) x^2-4 k x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(k-2) (1-x)^{2/3} x^{7/3}}{\sqrt [3]{1-k x} \left (-b \left (1-\frac {k^4}{b}\right ) x^4+2 b \left (1-\frac {2 k^3}{b}\right ) x^3-b \left (1-\frac {6 k^2}{b}\right ) x^2-4 k x+1\right )}+\frac {(1-x)^{2/3} x^{4/3}}{\sqrt [3]{1-k x} \left (-b \left (1-\frac {k^4}{b}\right ) x^4+2 b \left (1-\frac {2 k^3}{b}\right ) x^3-b \left (1-\frac {6 k^2}{b}\right ) x^2-4 k x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (\int \frac {(1-x)^{2/3} x^{4/3}}{\sqrt [3]{1-k x} \left (-b \left (1-\frac {k^4}{b}\right ) x^4+2 b \left (1-\frac {2 k^3}{b}\right ) x^3-b \left (1-\frac {6 k^2}{b}\right ) x^2-4 k x+1\right )}d\sqrt [3]{x}-(2-k) \int \frac {(1-x)^{2/3} x^{7/3}}{\sqrt [3]{1-k x} \left (-b \left (1-\frac {k^4}{b}\right ) x^4+2 b \left (1-\frac {2 k^3}{b}\right ) x^3-b \left (1-\frac {6 k^2}{b}\right ) x^2-4 k x+1\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Input:
Int[((-1 + x)*x*(1 + (-2 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4)),x]
Output:
$Aborted
\[\int \frac {\left (-1+x \right ) x \left (1+\left (-2+k \right ) x \right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-4 k x +\left (6 k^{2}-b \right ) x^{2}+\left (-4 k^{3}+2 b \right ) x^{3}+\left (k^{4}-b \right ) x^{4}\right )}d x\]
Input:
int((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b)*x^2+ (-4*k^3+2*b)*x^3+(k^4-b)*x^4),x)
Output:
int((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b)*x^2+ (-4*k^3+2*b)*x^3+(k^4-b)*x^4),x)
Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b )*x^2+(-4*k^3+2*b)*x^3+(k^4-b)*x^4),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))**(1/3)/(1-4*k*x+(6*k**2 -b)*x**2+(-4*k**3+2*b)*x**3+(k**4-b)*x**4),x)
Output:
Timed out
\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (k - 2\right )} x + 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b )*x^2+(-4*k^3+2*b)*x^3+(k^4-b)*x^4),x, algorithm="maxima")
Output:
integrate(((k - 2)*x + 1)*(x - 1)*x/(((k^4 - b)*x^4 - 2*(2*k^3 - b)*x^3 + (6*k^2 - b)*x^2 - 4*k*x + 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (k - 2\right )} x + 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b )*x^2+(-4*k^3+2*b)*x^3+(k^4-b)*x^4),x, algorithm="giac")
Output:
integrate(((k - 2)*x + 1)*(x - 1)*x/(((k^4 - b)*x^4 - 2*(2*k^3 - b)*x^3 + (6*k^2 - b)*x^2 - 4*k*x + 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=-\int \frac {x\,\left (x\,\left (k-2\right )+1\right )\,\left (x-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k^4\right )\,x^4+\left (4\,k^3-2\,b\right )\,x^3+\left (b-6\,k^2\right )\,x^2+4\,k\,x-1\right )} \,d x \] Input:
int(-(x*(x*(k - 2) + 1)*(x - 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(x^4*(b - k^ 4) + x^2*(b - 6*k^2) + 4*k*x - x^3*(2*b - 4*k^3) - 1)),x)
Output:
-int((x*(x*(k - 2) + 1)*(x - 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(x^4*(b - k^ 4) + x^2*(b - 6*k^2) + 4*k*x - x^3*(2*b - 4*k^3) - 1)), x)
\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx =\text {Too large to display} \] Input:
int((-1+x)*x*(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(1-4*k*x+(6*k^2-b)*x^2+ (-4*k^3+2*b)*x^3+(k^4-b)*x^4),x)
Output:
- int(x**3/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 - 2*x**(1/3)*(k *x**2 - k*x - x + 1)**(1/3)*b*x**3 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3 )*b*x**2 - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**4*x**4 + 4*x**(1/3)*( k*x**2 - k*x - x + 1)**(1/3)*k**3*x**3 - 6*x**(1/3)*(k*x**2 - k*x - x + 1) **(1/3)*k**2*x**2 + 4*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k*x - x**(1/3 )*(k*x**2 - k*x - x + 1)**(1/3)),x)*k + 2*int(x**3/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 - 2*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**3 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**2 - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**4*x**4 + 4*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**3*x **3 - 6*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**2*x**2 + 4*x**(1/3)*(k*x **2 - k*x - x + 1)**(1/3)*k*x - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)),x) + int(x**2/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 - 2*x**(1/3)*(k *x**2 - k*x - x + 1)**(1/3)*b*x**3 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3 )*b*x**2 - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**4*x**4 + 4*x**(1/3)*( k*x**2 - k*x - x + 1)**(1/3)*k**3*x**3 - 6*x**(1/3)*(k*x**2 - k*x - x + 1) **(1/3)*k**2*x**2 + 4*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k*x - x**(1/3 )*(k*x**2 - k*x - x + 1)**(1/3)),x)*k - 3*int(x**2/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 - 2*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**3 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**2 - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**4*x**4 + 4*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k**...