\(\int \frac {-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} (b-(1+2 b) x+(b+k) x^2)} \, dx\) [2561]

Optimal result

Integrand size = 52, antiderivative size = 217 \[ \int \frac {-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left (b-(1+2 b) x+(b+k) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2 x-2 k x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{b^{2/3}}+\frac {\log \left (-x+k x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{b^{2/3}}-\frac {\log \left (x^2-2 k x^3+k^2 x^4+\left (\sqrt [3]{b} x-\sqrt [3]{b} k x^2\right ) \left (x+(-1-k) x^2+k x^3\right )^{2/3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{4/3}\right )}{2 b^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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