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

Optimal. Leaf size=203 \[ \frac {(-a-b) \log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\sqrt [3]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+x^2\right )}{2 \sqrt [3]{b}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}+x}\right )}{\sqrt [3]{b}}+\frac {3 x^2}{2 \left (k x^3-k x^2-x^2+x\right )^{2/3}} \]

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Rubi [F]  time = 54.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 {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) \, _2F_1\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) \, _2F_1\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*}

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Mathematica [F]  time = 14.56, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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]

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]

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IntegrateAlgebraic [A]  time = 0.77, size = 203, normalized size = 1.00 \begin {gather*} \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 ) \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x +\left (1+k \right ) x^{2}\right ) \left (a -a \left (1+k \right ) x +\left (a k +1\right ) x^{2}\right )}{\left (-1+x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (k x -1\right ) \left (b -b \left (1+k \right ) x +\left (b k -1\right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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