\(\int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(b+2 k) x+(b+k^2) x^2)} \, dx\) [2469]

Optimal result

Integrand size = 48, antiderivative size = 201 \[ \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (-1+k x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (1-2 k x+k^2 x^2+\left (\sqrt [3]{b}-\sqrt [3]{b} k x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 15.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.80 \[ \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}{2-2 k x+\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (-1+k x+\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (1-2 k x+k^2 x^2+\sqrt [3]{b} (1-k x) \sqrt [3]{(-1+x) x (-1+k x)}+b^{2/3} ((-1+x) x (-1+k x))^{2/3}\right )}{2 b^{2/3}} \] Input:

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

Output:

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

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx=-\left (\int \frac {x}{x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b -x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b +x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k^{2}-2 x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k +x^{\frac {1}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}}}d x \right ) k +2 \left (\int \frac {x}{x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b -x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b +x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k^{2}-2 x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k +x^{\frac {1}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}}}d x \right )-\left (\int \frac {1}{x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b -x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} b +x^{\frac {7}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k^{2}-2 x^{\frac {4}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}} k +x^{\frac {1}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {1}{3}}}d x \right ) \] Input:

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

Output:

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