\(\int \frac {-1+2 x+(-2 k+k^2) x^2}{((1-x) x (1-k x))^{2/3} (b-(1+2 b k) x+(1+b k^2) x^2)} \, dx\) [2498]

Optimal result

Integrand size = 58, antiderivative size = 208 \[ \int \frac {-1+2 x+\left (-2 k+k^2\right ) x^2}{((1-x) x (1-k x))^{2/3} \left (b-(1+2 b k) x+\left (1+b k^2\right ) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2 x-2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{b^{2/3}}+\frac {\log \left (-x+x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{b^{2/3}}-\frac {\log \left (x^2-2 x^3+x^4+\left (\sqrt [3]{b} x-\sqrt [3]{b} x^2\right ) \left (x+(-1-k) x^2+k x^3\right )^{2/3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{4/3}\right )}{2 b^{2/3}} \]

[Out]

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

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 15.66 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.79 \[ \int \frac {-1+2 x+\left (-2 k+k^2\right ) x^2}{((1-x) x (1-k x))^{2/3} \left (b-(1+2 b k) x+\left (1+b k^2\right ) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} ((-1+x) x (-1+k x))^{2/3}}{2 x-2 x^2+\sqrt [3]{b} ((-1+x) x (-1+k x))^{2/3}}\right )+2 \log \left (-x+x^2+\sqrt [3]{b} ((-1+x) x (-1+k x))^{2/3}\right )-\log \left (x^2-2 x^3+x^4-\sqrt [3]{b} (-1+x) x ((-1+x) x (-1+k x))^{2/3}+b^{2/3} ((-1+x) x (-1+k x))^{4/3}\right )}{2 b^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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