\(\int \frac {-1+(-1+2 k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(2+b) x+(1+b k) x^2)} \, dx\) [2390]

Optimal result

Integrand size = 46, antiderivative size = 192 \[ \int \frac {-1+(-1+2 k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (-1+x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (1-2 x+x^2+\left (\sqrt [3]{b}-\sqrt [3]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-(((1 + Sqrt[4 + b - 4*k]/Sqrt[b] - 2*k)*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Int][1/((1 - k*x)^(1/3)*(
-2 - b - Sqrt[b]*Sqrt[4 + b - 4*k] + 2*(1 + b*k)*x)*(x - x^2)^(1/3)), x])/((1 - x)*x*(1 - k*x))^(1/3)) - ((1 -
Sqrt[4 + b - 4*k]/Sqrt[b] - 2*k)*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Int][1/((1 - k*x)^(1/3)*(-2 - b
+ Sqrt[b]*Sqrt[4 + b - 4*k] + 2*(1 + b*k)*x)*(x - x^2)^(1/3)), x])/((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+(-1+2 k) x}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (1-(2+b) x+(1+b k) x^2\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+(-1+2 k) x}{\sqrt [3]{1-k x} \sqrt [3]{x-x^2} \left (1-(2+b) x+(1+b k) x^2\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 \left (\frac {-1-\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k}{\sqrt [3]{1-k x} \left (-2-b-\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right ) \sqrt [3]{x-x^2}}+\frac {-1+\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k}{\sqrt [3]{1-k x} \left (-2-b+\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right ) \sqrt [3]{x-x^2}}\right ) \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (\left (-1-\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-k x} \left (-2-b-\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right ) \sqrt [3]{x-x^2}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (-1+\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-k x} \left (-2-b+\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right ) \sqrt [3]{x-x^2}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {-1+(-1+2 k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}{2-2 x+\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (-1+x+\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (1-2 x+x^2-\sqrt [3]{b} (-1+x) \sqrt [3]{(-1+x) x (-1+k x)}+b^{2/3} ((-1+x) x (-1+k x))^{2/3}\right )}{2 b^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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