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

Optimal. Leaf size=27 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a])

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Rubi [A]  time = 0.0496006, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*x^3]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a])

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Rubi in Sympy [A]  time = 5.23633, size = 26, normalized size = 0.96 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(a + b*x**3)/sqrt(a))/(3*sqrt(a))

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Mathematica [A]  time = 0.047194, size = 44, normalized size = 1.63 \[ -\frac{2 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 20, normalized size = 0.7 \[ -{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^(1/2),x)

[Out]

-2/3*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235079, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right )}{3 \, \sqrt{a}}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right )}{3 \, \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(b*x^3 + a)*a)/x^3)/sqrt(a), 2/3*arctan(
a/(sqrt(b*x^3 + a)*sqrt(-a)))/sqrt(-a)]

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Sympy [A]  time = 3.72377, size = 26, normalized size = 0.96 \[ - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(3*sqrt(a))

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GIAC/XCAS [A]  time = 0.257831, size = 31, normalized size = 1.15 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a)