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

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

[Out]

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

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Rubi [F]  time = 0.461683, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \]

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*}

Mathematica [F]  time = 0.668614, size = 0, normalized size = 0. \[ \int \frac{1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \]

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 = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{-kx+1}{1+ \left ( -2+k \right ) x} \left ( \left ( 1-x \right ) x \left ( -kx+1 \right ) \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} -\int \frac{k x - 1}{\left ({\left (k x - 1\right )}{\left (x - 1\right )} x\right )^{\frac{2}{3}}{\left ({\left (k - 2\right )} x + 1\right )}}\,{d x} \end{align*}

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

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]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{k x - 1}{\left ({\left (k x - 1\right )}{\left (x - 1\right )} x\right )^{\frac{2}{3}}{\left ({\left (k - 2\right )} x + 1\right )}}\,{d x} \end{align*}

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)