3.1.45 \(\int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx\) [45]

Optimal. Leaf size=176 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-k x)}{\sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-(2-k) x)}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-k x)}{2\ 2^{2/3} \sqrt [3]{1-k}}-\frac {3 \log \left (-1+k x+2^{2/3} \sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}\right )}{2\ 2^{2/3} \sqrt [3]{1-k}} \]

[Out]

1/2*ln(1-(2-k)*x)*2^(1/3)/(1-k)^(1/3)+1/4*ln(-k*x+1)*2^(1/3)/(1-k)^(1/3)-3/4*ln(-1+k*x+2^(2/3)*(1-k)^(1/3)*((1
-x)*x*(-k*x+1))^(1/3))*2^(1/3)/(1-k)^(1/3)-1/2*arctan(1/3*(1+2^(1/3)*(-k*x+1)/(1-k)^(1/3)/((1-x)*x*(-k*x+1))^(
1/3))*3^(1/2))*3^(1/2)*2^(1/3)/(1-k)^(1/3)

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Rubi [F]
time = 0.28, 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 {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]
time = 10.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-k x +1}{\left (1+\left (-2+k \right ) x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (132) = 264\).
time = 36.70, size = 932, normalized size = 5.30 \begin {gather*} \frac {\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\frac {24 \, \sqrt {3} 2^{\frac {1}{3}} {\left ({\left (k^{5} - 3 \, k^{4} - 4 \, k^{3} + 22 \, k^{2} - 24 \, k + 8\right )} x^{4} - 2 \, {\left (k^{4} - 10 \, k^{3} + 27 \, k^{2} - 26 \, k + 8\right )} x^{3} - 6 \, {\left (k^{3} - 4 \, k^{2} + 4 \, k - 1\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + k - 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}}}{{\left (k - 1\right )}^{\frac {1}{3}}} - \frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left ({\left (k^{6} + 27 \, k^{5} - 40 \, k^{4} - 20 \, k^{3} + 48 \, k^{2} - 16 \, k\right )} x^{5} - {\left (33 \, k^{5} + 55 \, k^{4} - 220 \, k^{3} + 132 \, k^{2} + 16 \, k - 16\right )} x^{4} + 2 \, {\left (55 \, k^{4} - 55 \, k^{3} - 66 \, k^{2} + 82 \, k - 16\right )} x^{3} - 2 \, {\left (55 \, k^{3} - 99 \, k^{2} + 38 \, k + 6\right )} x^{2} + {\left (33 \, k^{2} - 61 \, k + 28\right )} x - k + 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}}}{{\left (k - 1\right )}^{\frac {2}{3}}} + \sqrt {3} {\left ({\left (k^{6} - 48 \, k^{5} - 192 \, k^{4} + 416 \, k^{3} - 48 \, k^{2} - 192 \, k + 64\right )} x^{6} + 6 \, {\left (7 \, k^{5} + 104 \, k^{4} - 80 \, k^{3} - 176 \, k^{2} + 176 \, k - 32\right )} x^{5} - 3 \, {\left (139 \, k^{4} + 256 \, k^{3} - 768 \, k^{2} + 352 \, k + 16\right )} x^{4} + 4 \, {\left (203 \, k^{3} - 192 \, k^{2} - 120 \, k + 104\right )} x^{3} - 3 \, {\left (139 \, k^{2} - 208 \, k + 64\right )} x^{2} + 6 \, {\left (7 \, k - 8\right )} x + 1\right )}}{3 \, {\left ({\left (k^{6} + 96 \, k^{5} - 48 \, k^{4} - 160 \, k^{3} + 240 \, k^{2} - 192 \, k + 64\right )} x^{6} - 6 \, {\left (17 \, k^{5} + 64 \, k^{4} - 112 \, k^{3} + 80 \, k^{2} - 80 \, k + 32\right )} x^{5} + 3 \, {\left (149 \, k^{4} + 32 \, k^{3} - 96 \, k^{2} - 160 \, k + 80\right )} x^{4} - 4 \, {\left (157 \, k^{3} - 24 \, k^{2} - 168 \, k + 40\right )} x^{3} + 3 \, {\left (149 \, k^{2} - 128 \, k - 16\right )} x^{2} - 6 \, {\left (17 \, k - 16\right )} x + 1\right )}}\right )}{6 \, {\left (k - 1\right )}^{\frac {1}{3}}} - \frac {2^{\frac {1}{3}} \log \left (\frac {\frac {12 \cdot 2^{\frac {2}{3}} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}} {\left ({\left (k^{3} + k^{2} - 4 \, k + 2\right )} x^{2} - 2 \, {\left (2 \, k^{2} - 3 \, k + 1\right )} x + k - 1\right )}}{{\left (k - 1\right )}^{\frac {2}{3}}} + 6 \, {\left ({\left (k^{3} + 8 \, k^{2} - 8 \, k\right )} x^{3} - {\left (11 \, k^{2} - 8\right )} x^{2} + {\left (11 \, k - 8\right )} x - 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}} + \frac {2^{\frac {1}{3}} {\left ({\left (k^{4} + 28 \, k^{3} - 12 \, k^{2} - 32 \, k + 16\right )} x^{4} - 4 \, {\left (8 \, k^{3} + 15 \, k^{2} - 30 \, k + 8\right )} x^{3} + 6 \, {\left (13 \, k^{2} - 10 \, k - 2\right )} x^{2} - 4 \, {\left (8 \, k - 7\right )} x + 1\right )}}{{\left (k - 1\right )}^{\frac {1}{3}}}}{{\left (k^{4} - 8 \, k^{3} + 24 \, k^{2} - 32 \, k + 16\right )} x^{4} + 4 \, {\left (k^{3} - 6 \, k^{2} + 12 \, k - 8\right )} x^{3} + 6 \, {\left (k^{2} - 4 \, k + 4\right )} x^{2} + 4 \, {\left (k - 2\right )} x + 1}\right )}{12 \, {\left (k - 1\right )}^{\frac {1}{3}}} + \frac {2^{\frac {1}{3}} \log \left (\frac {\frac {6 \cdot 2^{\frac {1}{3}} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}} {\left (k x - 1\right )}}{{\left (k - 1\right )}^{\frac {1}{3}}} - \frac {2^{\frac {2}{3}} {\left ({\left (k^{2} - 4 \, k + 4\right )} x^{2} + 2 \, {\left (k - 2\right )} x + 1\right )}}{{\left (k - 1\right )}^{\frac {2}{3}}} - 12 \, {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}}}{{\left (k^{2} - 4 \, k + 4\right )} x^{2} + 2 \, {\left (k - 2\right )} x + 1}\right )}{6 \, {\left (k - 1\right )}^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*2^(1/3)*arctan(1/3*(24*sqrt(3)*2^(1/3)*((k^5 - 3*k^4 - 4*k^3 + 22*k^2 - 24*k + 8)*x^4 - 2*(k^4 - 1
0*k^3 + 27*k^2 - 26*k + 8)*x^3 - 6*(k^3 - 4*k^2 + 4*k - 1)*x^2 - 2*(k^2 - 1)*x + k - 1)*(k*x^3 - (k + 1)*x^2 +
x)^(2/3)/(k - 1)^(1/3) - 6*sqrt(3)*2^(2/3)*((k^6 + 27*k^5 - 40*k^4 - 20*k^3 + 48*k^2 - 16*k)*x^5 - (33*k^5 +
55*k^4 - 220*k^3 + 132*k^2 + 16*k - 16)*x^4 + 2*(55*k^4 - 55*k^3 - 66*k^2 + 82*k - 16)*x^3 - 2*(55*k^3 - 99*k^
2 + 38*k + 6)*x^2 + (33*k^2 - 61*k + 28)*x - k + 1)*(k*x^3 - (k + 1)*x^2 + x)^(1/3)/(k - 1)^(2/3) + sqrt(3)*((
k^6 - 48*k^5 - 192*k^4 + 416*k^3 - 48*k^2 - 192*k + 64)*x^6 + 6*(7*k^5 + 104*k^4 - 80*k^3 - 176*k^2 + 176*k -
32)*x^5 - 3*(139*k^4 + 256*k^3 - 768*k^2 + 352*k + 16)*x^4 + 4*(203*k^3 - 192*k^2 - 120*k + 104)*x^3 - 3*(139*
k^2 - 208*k + 64)*x^2 + 6*(7*k - 8)*x + 1))/((k^6 + 96*k^5 - 48*k^4 - 160*k^3 + 240*k^2 - 192*k + 64)*x^6 - 6*
(17*k^5 + 64*k^4 - 112*k^3 + 80*k^2 - 80*k + 32)*x^5 + 3*(149*k^4 + 32*k^3 - 96*k^2 - 160*k + 80)*x^4 - 4*(157
*k^3 - 24*k^2 - 168*k + 40)*x^3 + 3*(149*k^2 - 128*k - 16)*x^2 - 6*(17*k - 16)*x + 1))/(k - 1)^(1/3) - 1/12*2^
(1/3)*log((12*2^(2/3)*(k*x^3 - (k + 1)*x^2 + x)^(2/3)*((k^3 + k^2 - 4*k + 2)*x^2 - 2*(2*k^2 - 3*k + 1)*x + k -
1)/(k - 1)^(2/3) + 6*((k^3 + 8*k^2 - 8*k)*x^3 - (11*k^2 - 8)*x^2 + (11*k - 8)*x - 1)*(k*x^3 - (k + 1)*x^2 + x
)^(1/3) + 2^(1/3)*((k^4 + 28*k^3 - 12*k^2 - 32*k + 16)*x^4 - 4*(8*k^3 + 15*k^2 - 30*k + 8)*x^3 + 6*(13*k^2 - 1
0*k - 2)*x^2 - 4*(8*k - 7)*x + 1)/(k - 1)^(1/3))/((k^4 - 8*k^3 + 24*k^2 - 32*k + 16)*x^4 + 4*(k^3 - 6*k^2 + 12
*k - 8)*x^3 + 6*(k^2 - 4*k + 4)*x^2 + 4*(k - 2)*x + 1))/(k - 1)^(1/3) + 1/6*2^(1/3)*log((6*2^(1/3)*(k*x^3 - (k
+ 1)*x^2 + x)^(1/3)*(k*x - 1)/(k - 1)^(1/3) - 2^(2/3)*((k^2 - 4*k + 4)*x^2 + 2*(k - 2)*x + 1)/(k - 1)^(2/3) -
12*(k*x^3 - (k + 1)*x^2 + x)^(2/3))/((k^2 - 4*k + 4)*x^2 + 2*(k - 2)*x + 1))/(k - 1)^(1/3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {k x}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {k\,x-1}{\left (x\,\left (k-2\right )+1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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