\(\int \frac {(-1+(-1+2 k) x) (1-2 x+x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+(-b+k^2) x^4)} \, dx\) [2934]

Optimal result

Integrand size = 76, antiderivative size = 343 \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{b}}}{(-1+x)^2}\right )}{2 b^{2/3}}-\frac {\log \left (\sqrt [6]{b}-\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{2/3}}-\frac {\log \left (-\sqrt [6]{b}+\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2+\left (\sqrt [6]{b}-\sqrt [6]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{4 b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2+\left (-\sqrt [6]{b}+\sqrt [6]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{4 b^{2/3}} \]

[Out]

1/2*3^(1/2)*arctan((1/3*3^(1/2)-2/3*x*3^(1/2)+1/3*3^(1/2)*x^2+2/3*(x+(-1-k)*x^2+k*x^3)^(2/3)*3^(1/2)/b^(1/3))/
(-1+x)^2)/b^(2/3)-1/2*ln(b^(1/6)-b^(1/6)*x+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(2/3)-1/2*ln(-b^(1/6)+b^(1/6)*x+(x+(-
1-k)*x^2+k*x^3)^(1/3))/b^(2/3)+1/4*ln(b^(1/3)-2*b^(1/3)*x+b^(1/3)*x^2+(b^(1/6)-b^(1/6)*x)*(x+(-1-k)*x^2+k*x^3)
^(1/3)+(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(2/3)+1/4*ln(b^(1/3)-2*b^(1/3)*x+b^(1/3)*x^2+(-b^(1/6)+b^(1/6)*x)*(x+(-1-
k)*x^2+k*x^3)^(1/3)+(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(2/3)

Rubi [F]

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

[In]

Int[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*b - 2*
k)*x^3 + (-b + k^2)*x^4)),x]

[Out]

(3*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][(x*(1 - x^3)^(5/3))/((1 - k*x^3)^(1/3)*(b*(-1
+ x^3)^4 - x^6*(-1 + k*x^3)^2)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3) + (3*(1 - 2*k)*(1 - x)^(1/3)*x^
(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][(x^4*(1 - x^3)^(5/3))/((1 - k*x^3)^(1/3)*(b*(-1 + x^3)^4 - x^6*(
-1 + k*x^3)^2)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3)

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-1+x)^2 (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx \\ & = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(-1+x)^2 (-1+(-1+2 k) x)}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\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)^{5/3} (-1+(-1+2 k) x)}{\sqrt [3]{x} \sqrt [3]{1-k x} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\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 \left (1-x^3\right )^{5/3} \left (-1+(-1+2 k) x^3\right )}{\sqrt [3]{1-k x^3} \left (-b+4 b x^3+(1-6 b) x^6+(4 b-2 k) x^9+\left (-b+k^2\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 \frac {x \left (1-x^3\right )^{5/3} \left (1-(-1+2 k) x^3\right )}{\sqrt [3]{1-k x^3} \left (b \left (-1+x^3\right )^4-x^6 \left (-1+k x^3\right )^2\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 \left (1-x^3\right )^{5/3}}{\sqrt [3]{1-k x^3} \left (b-4 b x^3-(1-6 b) x^6-4 b \left (1-\frac {k}{2 b}\right ) x^9+b \left (1-\frac {k^2}{b}\right ) x^{12}\right )}+\frac {(1-2 k) x^4 \left (1-x^3\right )^{5/3}}{\sqrt [3]{1-k x^3} \left (b-4 b x^3-(1-6 b) x^6-4 b \left (1-\frac {k}{2 b}\right ) x^9+b \left (1-\frac {k^2}{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 \left (1-x^3\right )^{5/3}}{\sqrt [3]{1-k x^3} \left (b-4 b x^3-(1-6 b) x^6-4 b \left (1-\frac {k}{2 b}\right ) x^9+b \left (1-\frac {k^2}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (3 (1-2 k) \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 )^{5/3}}{\sqrt [3]{1-k x^3} \left (b-4 b x^3-(1-6 b) x^6-4 b \left (1-\frac {k}{2 b}\right ) x^9+b \left (1-\frac {k^2}{b}\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 \frac {x \left (1-x^3\right )^{5/3}}{\sqrt [3]{1-k x^3} \left (b \left (-1+x^3\right )^4-x^6 \left (-1+k x^3\right )^2\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (3 (1-2 k) \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 )^{5/3}}{\sqrt [3]{1-k x^3} \left (b \left (-1+x^3\right )^4-x^6 \left (-1+k x^3\right )^2\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 25.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.79 \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\frac {\sqrt [3]{k-\frac {1}{x}} (-1+x) \left (-2 \sqrt {3} \arctan \left (\frac {\left (k-\frac {1}{x}\right )^{2/3}+2 \sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}}{\sqrt {3} \left (k-\frac {1}{x}\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}-\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}+\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}-\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}+\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )\right )}{4 b^{2/3} \left (-1+\frac {1}{x}\right )^{2/3} \sqrt [3]{(-1+x) x (-1+k x)}} \]

[In]

Integrate[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*
b - 2*k)*x^3 + (-b + k^2)*x^4)),x]

[Out]

((k - x^(-1))^(1/3)*(-1 + x)*(-2*Sqrt[3]*ArcTan[((k - x^(-1))^(2/3) + 2*b^(1/3)*(-1 + x^(-1))^(4/3))/(Sqrt[3]*
(k - x^(-1))^(2/3))] - 2*Log[(k - x^(-1))^(1/3) - b^(1/6)*(-1 + x^(-1))^(2/3)] - 2*Log[(k - x^(-1))^(1/3) + b^
(1/6)*(-1 + x^(-1))^(2/3)] + Log[(k - x^(-1))^(2/3) - b^(1/6)*(k - x^(-1))^(1/3)*(-1 + x^(-1))^(2/3) + b^(1/3)
*(-1 + x^(-1))^(4/3)] + Log[(k - x^(-1))^(2/3) + b^(1/6)*(k - x^(-1))^(1/3)*(-1 + x^(-1))^(2/3) + b^(1/3)*(-1
+ x^(-1))^(4/3)]))/(4*b^(2/3)*(-1 + x^(-1))^(2/3)*((-1 + x)*x*(-1 + k*x))^(1/3))

Maple [F]

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

[In]

int((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x)

[Out]

int((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x)

Fricas [F(-1)]

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

[In]

integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4
),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((-1+(-1+2*k)*x)*(x**2-2*x+1)/((1-x)*x*(-k*x+1))**(1/3)/(-b+4*b*x+(1-6*b)*x**2+(4*b-2*k)*x**3+(k**2-b
)*x**4),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4
),x, algorithm="maxima")

[Out]

integrate(((2*k - 1)*x - 1)*(x^2 - 2*x + 1)/(((k^2 - b)*x^4 + 2*(2*b - k)*x^3 - (6*b - 1)*x^2 + 4*b*x - b)*((k
*x - 1)*(x - 1)*x)^(1/3)), x)

Giac [F]

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

[In]

integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4
),x, algorithm="giac")

[Out]

integrate(((2*k - 1)*x - 1)*(x^2 - 2*x + 1)/(((k^2 - b)*x^4 + 2*(2*b - k)*x^3 - (6*b - 1)*x^2 + 4*b*x - b)*((k
*x - 1)*(x - 1)*x)^(1/3)), x)

Mupad [F(-1)]

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

[In]

int(-((x*(2*k - 1) - 1)*(x^2 - 2*x + 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x^3*(4*b - 2*k) + x^4*(b - k^2) - 4
*b*x + x^2*(6*b - 1))),x)

[Out]

int(-((x*(2*k - 1) - 1)*(x^2 - 2*x + 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x^3*(4*b - 2*k) + x^4*(b - k^2) - 4
*b*x + x^2*(6*b - 1))), x)