3.547 \(\int \frac{1}{x \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=83 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.118945, antiderivative size = 83, 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 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)^(1/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.28029, size = 78, normalized size = 0.94 \[ - \frac{\log{\left (x^{3} \right )}}{6 \sqrt [3]{a}} + \frac{\log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{2 \sqrt [3]{a}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0344027, size = 46, normalized size = 0.55 \[ -\frac{\sqrt [3]{\frac{a}{b x^3}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^3}\right )}{\sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-(((1 + a/(b*x^3))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(b*x^3))])/(a + b*
x^3)^(1/3))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^(1/3),x)

[Out]

int(1/x/(b*x^3+a)^(1/3),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)^(1/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252302, size = 130, normalized size = 1.57 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) - 2 \, \sqrt{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, a^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.70844, size = 37, normalized size = 0.45 \[ - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(1/3)*hyper((1/3, 1/3), (4/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b**(1/3)*x*
gamma(4/3))

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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)^(1/3)*x),x, algorithm="giac")

[Out]

Timed out