\(\int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} (b-(1+2 b k) x+(1+b k^2) x^2)} \, dx\) [2542]

Optimal result

Integrand size = 51, antiderivative size = 214 \[ \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (b-(1+2 b k) x+\left (1+b k^2\right ) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 \sqrt [3]{b}-2 \sqrt [3]{b} k x+\sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b}+\sqrt [3]{b} k x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b^{2/3}-2 b^{2/3} k x+b^{2/3} k^2 x^2+\left (\sqrt [3]{b}-\sqrt [3]{b} k x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]

[Out]

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

Rubi [F]

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

[In]

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

Mathematica [A] (verified)

Time = 15.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.71 \[ \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (b-(1+2 b k) x+\left (1+b k^2\right ) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{(-1+x) x (-1+k x)}}{\sqrt [3]{b} (2-2 k x)+\sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (\sqrt [3]{b} (-1+k x)+\sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (b^{2/3} (-1+k x)^2+\sqrt [3]{b} (1-k x) \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 \sqrt [3]{b}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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