3.44 \(\int \frac{2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx\)

Optimal. Leaf size=111 \[ \frac{\log (x)}{2 \sqrt [3]{k}}+\frac{\log (1-(k+1) x)}{2 \sqrt [3]{k}}-\frac{3 \log \left (\sqrt [3]{(1-x) x (1-k x)}-\sqrt [3]{k} x\right )}{2 \sqrt [3]{k}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{k} x}{\sqrt [3]{(1-x) x (1-k x)}}+1}{\sqrt{3}}\right )}{\sqrt [3]{k}} \]

[Out]

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

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Rubi [F]  time = 0.610747, 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 = {} \[ \int \frac{2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(3*(1 - x)^(1/3)*x*(1 - k*x)^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, x, k*x])/(2*((1 - x)*x*(1 - k*x))^(1/3)) + ((1
- x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Int][1/((1 - x)^(1/3)*x^(1/3)*(1 + (-1 - k)*x)*(1 - k*x)^(1/3)), x])
/((1 - x)*x*(1 - k*x))^(1/3)

Rubi steps

\begin{align*} \int \frac{2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx &=\frac{\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac{2-(1+k) x}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} (1-(1+k) x)} \, 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{1}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}} \, 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{1}{\sqrt [3]{1-x} \sqrt [3]{x} (1+(-1-k) x) \sqrt [3]{1-k x}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac{3 \sqrt [3]{1-x} x \sqrt [3]{1-k x} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};x,k x\right )}{2 \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{1}{\sqrt [3]{1-x} \sqrt [3]{x} (1+(-1-k) x) \sqrt [3]{1-k x}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ \end{align*}

Mathematica [F]  time = 1.63514, size = 0, normalized size = 0. \[ \int \frac{2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{2- \left ( 1+k \right ) x}{1- \left ( 1+k \right ) x}{\frac{1}{\sqrt [3]{ \left ( 1-x \right ) x \left ( -kx+1 \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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