\(\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} (1-4 k x+(-b+6 k^2) x^2+(2 b-4 k^3) x^3+(-b+k^4) x^4)} \, dx\) [2837]

Optimal result

Integrand size = 74, antiderivative size = 289 \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-2+2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-1+k x}\right )}{b^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (-\sqrt [6]{b}+\sqrt [6]{b} k x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}}{1-2 k x+k^2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{5/6}} \]

[Out]

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)

Rubi [F]

\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \]

[In]

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]

[Out]

(-3*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][(x^4*(1 - x^3)^(2/3))/((1 - k*x^3)^(1/3)*(1
- 4*k*x^3 - b*(1 - (6*k^2)/b)*x^6 + 2*b*(1 - (2*k^3)/b)*x^9 - b*(1 - k^4/b)*x^12)), x], x, x^(1/3)])/((1 - x)*
x*(1 - k*x))^(1/3) + (3*(2 - k)*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][(x^7*(1 - x^3)^(
2/3))/((1 - k*x^3)^(1/3)*(1 - 4*k*x^3 - b*(1 - (6*k^2)/b)*x^6 + 2*b*(1 - (2*k^3)/b)*x^9 - b*(1 - k^4/b)*x^12))
, x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(-1+x) x^{2/3} (1+(-2+k) x)}{\sqrt [3]{1-x} \sqrt [3]{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}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(1-x)^{2/3} x^{2/3} (1+(-2+k) x)}{\sqrt [3]{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}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-x^3\right )^{2/3} \left (1+(-2+k) x^3\right )}{\sqrt [3]{1-k x^3} \left (1-4 k x^3+\left (-b+6 k^2\right ) x^6+\left (2 b-4 k^3\right ) x^9+\left (-b+k^4\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \left (\frac {x^4 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}\right )}+\frac {(-2+k) x^7 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (3 (-2+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \]

[In]

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]

[Out]

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]

Maple [F]

\[\int \frac {\left (-1+x \right ) x \left (1+\left (-2+k \right ) x \right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-4 k x +\left (6 k^{2}-b \right ) x^{2}+\left (-4 k^{3}+2 b \right ) x^{3}+\left (k^{4}-b \right ) x^{4}\right )}d x\]

[In]

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)

[Out]

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)

Fricas [F(-1)]

Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \]

[In]

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")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (k - 2\right )} x + 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

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")

[Out]

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)

Giac [F]

\[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (k - 2\right )} x + 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=-\int \frac {x\,\left (x\,\left (k-2\right )+1\right )\,\left (x-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k^4\right )\,x^4+\left (4\,k^3-2\,b\right )\,x^3+\left (b-6\,k^2\right )\,x^2+4\,k\,x-1\right )} \,d x \]

[In]

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)

[Out]

-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)