3.43 \(\int \frac{1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{3 \log \left (\sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}-\sqrt [3]{q} (x-1)\right )}{4 \sqrt [3]{q}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{q} (x-1)}{\sqrt{3} \sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{q}}+\frac{\log (1-x)}{4 \sqrt [3]{q}}+\frac{\log (x)}{2 \sqrt [3]{q}} \]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*q^(1/3)*(-1 + x))/(Sqrt[3]*((-1 + x)*(q - 2*q*x +
x^2))^(1/3))])/(2*q^(1/3)) + Log[1 - x]/(4*q^(1/3)) + Log[x]/(2*q^(1/3)) - (3*L
og[-(q^(1/3)*(-1 + x)) + ((-1 + x)*(q - 2*q*x + x^2))^(1/3)])/(4*q^(1/3))

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Rubi [F]  time = 27.6796, 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. \[ \text{Int}\left (\frac{1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

((-1 - 2*q - (1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 -
q)^3*q])^(2/3))/(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(1/3) +
3*x)^(1/3)*(-1 + 5*q - 4*q^2 + ((1 - 4*q)^2*(1 - q)^2)/(1 + 6*q - 15*q^2 + 8*q^
3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3) + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*S
qrt[(1 - q)^3*q])^(2/3) - ((1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt
[3]*Sqrt[(1 - q)^3*q])^(2/3))*(1 + 2*q - 3*x))/(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqr
t[3]*Sqrt[(1 - q)^3*q])^(1/3) + (-1 - 2*q + 3*x)^2)^(1/3)*Defer[Subst][Defer[Int
][1/(((1 + 2*q)/3 + x)*(-(1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3
]*Sqrt[(1 - q)^3*q])^(2/3))/(3*(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q
)^3*q])^(1/3)) + x)^(1/3)*((-1 + 5*q - 4*q^2 + ((1 - 4*q)^2*(1 - q)^2)/(1 + 6*q
- 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3) + (1 + 6*q - 15*q^2 + 8*q^
3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3))/9 + ((1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^
2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3))*x)/(3*(1 + 6*q - 15*q^2 + 8*q^3
+ 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(1/3)) + x^2)^(1/3)), x], x, (-1 - 2*q)/3 + x])/(
3*(-q + 3*q*x - (1 + 2*q)*x^2 + x^3)^(1/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{x - 1} \sqrt [3]{- 2 q x + q + x^{2}} \int \frac{1}{x \sqrt [3]{x - 1} \sqrt [3]{- 2 q x + q + x^{2}}}\, dx}{\sqrt [3]{3 q x - q + x^{3} + x^{2} \left (- 2 q - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/((-1+x)*(-2*q*x+x**2+q))**(1/3),x)

[Out]

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

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Mathematica [C]  time = 0.22792, size = 72, normalized size = 0.61 \[ \frac{3 (x-1) \sqrt [3]{-\frac{-2 q x+q+x^2}{(q-1) x^2}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{q (x-1)^2}{(q-1) x^2}\right )}{2 \sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-1 + x)*(-((q - 2*q*x + x^2)/((-1 + q)*x^2)))^(1/3)*Hypergeometric2F1[1/3, 1
/3, 4/3, (q*(-1 + x)^2)/((-1 + q)*x^2)])/(2*((-1 + x)*(q - 2*q*x + x^2))^(1/3))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{ \left ( -1+x \right ) \left ( -2\,qx+{x}^{2}+q \right ) }}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (-{\left (2 \, q x - x^{2} - q\right )}{\left (x - 1\right )}\right )^{\frac{1}{3}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/((-1+x)*(-2*q*x+x**2+q))**(1/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (-{\left (2 \, q x - x^{2} - q\right )}{\left (x - 1\right )}\right )^{\frac{1}{3}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x),x, algorithm="giac")

[Out]

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