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

Optimal. Leaf size=66 \[ -\frac{3}{4} \log \left (\sqrt [3]{x \left (x^2-q\right )}-x\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x}{\sqrt{3} \sqrt [3]{x \left (x^2-q\right )}}+\frac{1}{\sqrt{3}}\right )+\frac{\log (x)}{4} \]

[Out]

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

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Rubi [A]  time = 0.112684, antiderivative size = 117, normalized size of antiderivative = 1.77, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{\sqrt{3} \sqrt [3]{x} \sqrt [3]{x^2-q} \tan ^{-1}\left (\frac{\frac{2 x^{2/3}}{\sqrt [3]{x^2-q}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{x^3-q x}}-\frac{3 \sqrt [3]{x} \sqrt [3]{x^2-q} \log \left (x^{2/3}-\sqrt [3]{x^2-q}\right )}{4 \sqrt [3]{x^3-q x}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(-q + x^2))^(-1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.45697, size = 155, normalized size = 2.35 \[ - \frac{\left (- q x + x^{3}\right )^{\frac{2}{3}} \log{\left (- \frac{x^{\frac{2}{3}}}{\sqrt [3]{- q + x^{2}}} + 1 \right )}}{2 x^{\frac{2}{3}} \left (- q + x^{2}\right )^{\frac{2}{3}}} + \frac{\left (- q x + x^{3}\right )^{\frac{2}{3}} \log{\left (\frac{x^{\frac{4}{3}}}{\left (- q + x^{2}\right )^{\frac{2}{3}}} + \frac{x^{\frac{2}{3}}}{\sqrt [3]{- q + x^{2}}} + 1 \right )}}{4 x^{\frac{2}{3}} \left (- q + x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \left (- q x + x^{3}\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{\frac{2}{3}}}{3 \sqrt [3]{- q + x^{2}}} + \frac{1}{3}\right ) \right )}}{2 x^{\frac{2}{3}} \left (- q + x^{2}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-q*x + x**3)**(2/3)*log(-x**(2/3)/(-q + x**2)**(1/3) + 1)/(2*x**(2/3)*(-q + x*
*2)**(2/3)) + (-q*x + x**3)**(2/3)*log(x**(4/3)/(-q + x**2)**(2/3) + x**(2/3)/(-
q + x**2)**(1/3) + 1)/(4*x**(2/3)*(-q + x**2)**(2/3)) + sqrt(3)*(-q*x + x**3)**(
2/3)*atan(sqrt(3)*(2*x**(2/3)/(3*(-q + x**2)**(1/3)) + 1/3))/(2*x**(2/3)*(-q + x
**2)**(2/3))

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Mathematica [C]  time = 0.0347719, size = 49, normalized size = 0.74 \[ \frac{3 x \sqrt [3]{\frac{q-x^2}{q}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{x^2}{q}\right )}{2 \sqrt [3]{x^3-q x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{x \left ({x}^{2}-q \right ) }}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x*(x^2-q))^(1/3),x)

[Out]

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

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Maxima [A]  time = 1.52352, size = 104, normalized size = 1.58 \[ -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (x^{2} - q\right )}^{\frac{1}{3}}}{x^{\frac{2}{3}}} + 1\right )}\right ) + \frac{1}{4} \, \log \left (\frac{{\left (x^{2} - q\right )}^{\frac{1}{3}}}{x^{\frac{2}{3}}} + \frac{{\left (x^{2} - q\right )}^{\frac{2}{3}}}{x^{\frac{4}{3}}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{{\left (x^{2} - q\right )}^{\frac{1}{3}}}{x^{\frac{2}{3}}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^2 - q)^(1/3)/x^(2/3) + 1)) + 1/4*log((x^2
- q)^(1/3)/x^(2/3) + (x^2 - q)^(2/3)/x^(4/3) + 1) - 1/2*log((x^2 - q)^(1/3)/x^(2
/3) - 1)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(-q + x**2))**(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((x^2 - q)*x)^(-1/3), x)