\(\int \frac {(-2+(1+k) x) (a-a (1+k) x+(1+a k) x^2)}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) (b-b (1+k) x+(-1+b k) x^2)} \, dx\) [2528]

Optimal result

Integrand size = 75, antiderivative size = 212 \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 \left (x-x^2-k x^2+k x^3\right )^{2/3}}{(-1+x) (-1+k x)}+\frac {\left (-\sqrt {3} a-\sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}+\frac {(-a-b) \log \left (x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]

[Out]

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

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

[In]

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

[Out]

(-3*(1 + k)*(1 + a*k)*x)/((1 - k)*(1 - b*k)*((1 - x)*x*(1 - k*x))^(1/3)) + (3*(2 + a + b + 4*a*k + a*(1 - 2*b)
*k^2 + b*k^2)*x)/((1 - k)*(1 - b*k)^2*((1 - x)*x*(1 - k*x))^(1/3)) - (3*(1 + k)*(1 + a*k)*(1 - x)*(((1 - k)*x)
/(1 - k*x))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (1 - x)/(1 - k*x)])/((1 - k)^2*(1 - b*k)*((1 - x)*x*(1 - k*
x))^(1/3)) + (3*(1 + k)*(2 + a + b + 4*a*k + a*(1 - 2*b)*k^2 + b*k^2)*(1 - x)*(((1 - k)*x)/(1 - k*x))^(1/3)*Hy
pergeometric2F1[1/3, 2/3, 5/3, (1 - x)/(1 - k*x)])/(2*(1 - k)^2*(1 - b*k)^2*((1 - x)*x*(1 - k*x))^(1/3)) + ((a
+ b)*(3 + b + 3*k + b*k^3 + (4 + b^2*(1 - k)^2*(1 + k + k^2) + b*(5 + 2*k + 5*k^2))/(Sqrt[b]*Sqrt[4 + b*(1 -
k)^2]))*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Int][1/((1 - x)^(4/3)*x^(1/3)*(1 - k*x)^(4/3)*(-(b*(1 + k)
) - Sqrt[b]*Sqrt[4 + b - 2*b*k + b*k^2] + 2*(-1 + b*k)*x)), x])/((1 - b*k)^2*((1 - x)*x*(1 - k*x))^(1/3)) + ((
a + b)*(3*(1 + k) + b*(1 + k^3) - (4 + b*(5 + 2*k + 5*k^2) + b^2*(1 - k - k^3 + k^4))/(Sqrt[b]*Sqrt[4 + b*(1 -
k)^2]))*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Int][1/((1 - x)^(4/3)*x^(1/3)*(1 - k*x)^(4/3)*(-(b*(1 + k
)) + Sqrt[b]*Sqrt[4 + b - 2*b*k + b*k^2] + 2*(-1 + b*k)*x)), x])/((1 - b*k)^2*((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 {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{\sqrt [3]{1-x} (-1+x) \sqrt [3]{x} \sqrt [3]{1-k x} (-1+k x) \left (b-b (1+k) 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 {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{4/3} \sqrt [3]{x} \sqrt [3]{1-k x} (-1+k x) \left (b-b (1+k) 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 {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (b-b (1+k) 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 {2+a+b+4 a k+a (1-2 b) k^2+b k^2}{(1-b k)^2 (1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3}}-\frac {(1+k) (1+a k) x^{2/3}}{(1-b k) (1-x)^{4/3} (1-k x)^{4/3}}-\frac {(a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x}{(-1+b k)^2 (1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (b-b (1+k) x+(-1+b k) x^2\right )}\right ) \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left ((1+k) (1+a k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {x^{2/3}}{(1-x)^{4/3} (1-k x)^{4/3}} \, dx}{(1-b k) \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 {(a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx}{(-1+b k)^2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3}} \, dx}{(1-b k)^2 \sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {3 (1+k) (1+a k) x}{(1-k) (1-b k) \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x}{(1-k) (1-b k)^2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (2 (1+k) (1+a k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} (1-k x)^{4/3}} \, dx}{(1-k) (1-b k) \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 {-\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )}+\frac {\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (-b (1+k)+\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )}\right ) \, dx}{(-1+b k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left ((1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} (1-k x)^{4/3}} \, dx}{(1-k) (1-b k)^2 \sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {3 (1+k) (1+a k) x}{(1-k) (1-b k) \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x}{(1-k) (1-b k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) (1+a k) (1-x) \sqrt [3]{\frac {(1-k) x}{1-k x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {1-x}{1-k x}\right )}{(1-k)^2 (1-b k) \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) \sqrt [3]{\frac {(1-k) x}{1-k x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {1-x}{1-k x}\right )}{2 (1-k)^2 (1-b k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (-b (1+k)+\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((a+b) \left (3+b+3 k+b k^3+\frac {4+b^2 (1-k)^2 \left (1+k+k^2\right )+b \left (5+2 k+5 k^2\right )}{\sqrt {b} \sqrt {4+b (1-k)^2}}\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{(1-x)^{4/3} \sqrt [3]{x} (1-k x)^{4/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 \sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 29.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.14 \[ \int \frac {(-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) \sqrt [3]{(1-x) x (1-k x)} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {(-1+x) \left (\frac {6 x}{-1+x}+\frac {(a+b) \sqrt [3]{\frac {x}{-1+x}} \sqrt [3]{\frac {-1+k x}{-1+x}} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}}{2 \left (\frac {x}{-1+x}\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}}\right )+2 \log \left (\left (\frac {x}{-1+x}\right )^{2/3}-\sqrt [3]{b} \sqrt [3]{\frac {-1+k x}{-1+x}}\right )-\log \left (\left (\frac {x}{-1+x}\right )^{4/3}+\sqrt [3]{b} \left (\frac {x}{-1+x}\right )^{2/3} \sqrt [3]{\frac {-1+k x}{-1+x}}+b^{2/3} \left (\frac {-1+k x}{-1+x}\right )^{2/3}\right )\right )}{b^{2/3}}\right )}{2 \sqrt [3]{(-1+x) x (-1+k x)}} \]

[In]

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

[Out]

((-1 + x)*((6*x)/(-1 + x) + ((a + b)*(x/(-1 + x))^(1/3)*((-1 + k*x)/(-1 + x))^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]
*b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3))/(2*(x/(-1 + x))^(2/3) + b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3))] + 2*Log[(x
/(-1 + x))^(2/3) - b^(1/3)*((-1 + k*x)/(-1 + x))^(1/3)] - Log[(x/(-1 + x))^(4/3) + b^(1/3)*(x/(-1 + x))^(2/3)*
((-1 + k*x)/(-1 + x))^(1/3) + b^(2/3)*((-1 + k*x)/(-1 + x))^(2/3)]))/b^(2/3)))/(2*((-1 + x)*x*(-1 + k*x))^(1/3
))

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(-\frac {\left (a +b \right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{b}\right )^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{b}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {1}{b}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{b}\right )^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-\left (\frac {1}{b}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )\right ) \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}-6 x b \left (\frac {1}{b}\right )^{\frac {1}{3}}}{2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} \left (\frac {1}{b}\right )^{\frac {1}{3}} b}\)	\(165\)

[In]

int((-2+(1+k)*x)*(a-a*(1+k)*x+(a*k+1)*x^2)/(-1+x)/((1-x)*x*(-k*x+1))^(1/3)/(k*x-1)/(b-b*(1+k)*x+(b*k-1)*x^2),x
,method=_RETURNVERBOSE)

[Out]

-1/2*((a+b)*(-2*arctan(1/3*3^(1/2)*((1/b)^(1/3)*x+2*((-1+x)*x*(k*x-1))^(1/3))/(1/b)^(1/3)/x)*3^(1/2)+ln(((1/b)
^(2/3)*x^2+(1/b)^(1/3)*((-1+x)*x*(k*x-1))^(1/3)*x+((-1+x)*x*(k*x-1))^(2/3))/x^2)-2*ln((-(1/b)^(1/3)*x+((-1+x)*
x*(k*x-1))^(1/3))/x))*((-1+x)*x*(k*x-1))^(1/3)-6*x*b*(1/b)^(1/3))/((-1+x)*x*(k*x-1))^(1/3)/(1/b)^(1/3)/b

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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