3.564 \(\int \frac{1}{x \left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=84 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.115965, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.38794, size = 78, normalized size = 0.93 \[ - \frac{\log{\left (x^{3} \right )}}{6 a^{\frac{2}{3}}} + \frac{\log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{2 a^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**3+a)**(2/3),x)

[Out]

-log(x**3)/(6*a**(2/3)) + log(a**(1/3) - (a + b*x**3)**(1/3))/(2*a**(2/3)) - sqr
t(3)*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**3)**(1/3)/3)/a**(1/3))/(3*a**(2/3))

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Mathematica [C]  time = 0.0334808, size = 48, normalized size = 0.57 \[ -\frac{\left (\frac{a}{b x^3}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^3}\right )}{2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^3)^(2/3)),x]

[Out]

-((1 + a/(b*x^3))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -(a/(b*x^3))])/(2*(a +
b*x^3)^(2/3))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^(2/3),x)

[Out]

int(1/x/(b*x^3+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251581, size = 147, normalized size = 1.75 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (a^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \log \left (-a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}\right ) + 6 \, \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}}{3 \, a}\right )\right )}}{18 \,{\left (a^{2}\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*log(a^2 + (b*x^3 + a)^(1/3)*(a^2)^(1/3)*a + (b*x^3 + a)^(
2/3)*(a^2)^(2/3)) - 2*sqrt(3)*log(-a + (b*x^3 + a)^(1/3)*(a^2)^(1/3)) + 6*arctan
(1/3*(sqrt(3)*a + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(a^2)^(1/3))/a))/(a^2)^(1/3)

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Sympy [A]  time = 3.86497, size = 39, normalized size = 0.46 \[ - \frac{\Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 b^{\frac{2}{3}} x^{2} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**3+a)**(2/3),x)

[Out]

-gamma(2/3)*hyper((2/3, 2/3), (5/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b**(2/3)*x*
*2*gamma(5/3))

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GIAC/XCAS [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/((b*x^3 + a)^(2/3)*x),x, algorithm="giac")

[Out]

Timed out