\(\int \frac {(-2 x+(1+k) x^2) (a-a (1+k) x+(1+a k) x^2)}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) (b-b (1+k) x+(-1+b k) x^2)} \, dx\) [2472]

Optimal result

Integrand size = 79, antiderivative size = 203 \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 x^2}{2 \left (x-x^2-k x^2+k x^3\right )^{2/3}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {(-a-b) \log \left (x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-3*(1 + k)*(1 + a*k)*(1 - x)*x)/((1 - k)^2*(1 - b*k)*((1 - x)*x*(1 - k*x))^(2/3)) + (3*(1 + k)*(2 + a + b + 4
*a*k + a*(1 - 2*b)*k^2 + b*k^2)*(1 - x)*x)/(2*(1 - k)^2*(1 - b*k)^2*((1 - x)*x*(1 - k*x))^(2/3)) - (3*(1 + k)*
(1 + a*k)*x^2)/(2*(1 - k)*(1 - b*k)*((1 - x)*x*(1 - k*x))^(2/3)) + (3*(2 + a + b + 4*a*k + a*(1 - 2*b)*k^2 + b
*k^2)*x^2)/(2*(1 - k)*(1 - b*k)^2*((1 - x)*x*(1 - k*x))^(2/3)) - (3*(1 + k)*(1 + a*k)*(1 - x)*(((1 - k)*x)/(1
- k*x))^(2/3)*(1 - k*x)*Hypergeometric2F1[1/3, 2/3, 4/3, (1 - x)/(1 - k*x)])/((1 - k)^3*(1 - b*k)*((1 - x)*x*(
1 - k*x))^(2/3)) + (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))^(2
/3)*(1 - k*x)*Hypergeometric2F1[1/3, 2/3, 4/3, (1 - x)/(1 - k*x)])/(2*(1 - k)^3*(1 - b*k)^2*((1 - x)*x*(1 - k*
x))^(2/3)) + ((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)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Int][x^(1/3)/((1 - x)^(5/3)*(1 - k*x)^(5/3
)*(-(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
))^(2/3)) + ((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]*S
qrt[4 + b*(1 - k)^2]))*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Int][x^(1/3)/((1 - x)^(5/3)*(1 - k*x)^(5/3)
*(-(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)
)^(2/3))

Rubi steps \begin{align*} \text {integral}& = \int \frac {x (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx \\ & = \frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{2/3} (-1+x) (1-k x)^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+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]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{5/3} (1-k x)^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+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]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) x+(-1+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 (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}-\frac {(1+k) (1+a k) x^{4/3}}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\sqrt [3]{x} \left ((a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x\right )}{(-1+b k)^2 (1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) x+(-1+b k) x^2\right )}\right ) \, dx}{((1-x) x (1-k x))^{2/3}} \\ & = -\frac {\left ((1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {x^{4/3}}{(1-x)^{5/3} (1-k x)^{5/3}} \, dx}{(1-b k) ((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]{x} \left ((a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left (\left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3}} \, dx}{(1-b k)^2 ((1-x) x (1-k x))^{2/3}} \\ & = -\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left (2 (1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{2/3} (1-k x)^{5/3}} \, dx}{(1-k) (1-b k) ((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 (-\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]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )}+\frac {\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]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \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 ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{2/3} (1-k x)^{5/3}} \, dx}{(1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}} \\ & = -\frac {3 (1+k) (1+a k) (1-x) x}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) x}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\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 ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \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 ((1-x) x (1-k x))^{2/3}}+\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 ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \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 ((1-x) x (1-k x))^{2/3}} \\ & = -\frac {3 (1+k) (1+a k) (1-x) x}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) x}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {1-x}{1-k x}\right )}{(1-k)^3 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {1-x}{1-k x}\right )}{2 (1-k)^3 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\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 ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \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 ((1-x) x (1-k x))^{2/3}}+\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 ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \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 ((1-x) x (1-k x))^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 34.46 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx=\frac {3 x^2}{2 ((-1+x) x (-1+k x))^{2/3}}+\frac {(a+b) \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (x-\sqrt [3]{b} \sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+b^{2/3} ((-1+x) x (-1+k x))^{2/3}\right )\right )}{2 \sqrt [3]{b}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\frac {-\frac {3 x^{2} b \left (\frac {1}{b}\right )^{\frac {2}{3}}}{2}+\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (a +b \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{b}\right )^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{b}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\ln \left (\frac {-\left (\frac {1}{b}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {\left (\frac {1}{b}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{b}\right )^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} b}\)	\(168\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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