3.2105 \(\int \frac{1}{\sqrt{a+\frac{b}{x^5}} x} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

[Out]

(2*ArcTanh[Sqrt[a + b/x^5]/Sqrt[a]])/(5*Sqrt[a])

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Rubi [A]  time = 0.0624642, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^5]*x),x]

[Out]

(2*ArcTanh[Sqrt[a + b/x^5]/Sqrt[a]])/(5*Sqrt[a])

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Rubi in Sympy [A]  time = 5.12292, size = 24, normalized size = 0.89 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{5}}}}{\sqrt{a}} \right )}}{5 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b/x**5)**(1/2),x)

[Out]

2*atanh(sqrt(a + b/x**5)/sqrt(a))/(5*sqrt(a))

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Mathematica [A]  time = 0.0519659, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+\frac{b}{x^5}} x} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(Sqrt[a + b/x^5]*x),x]

[Out]

Integrate[1/(Sqrt[a + b/x^5]*x), x]

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt{a+{\frac{b}{{x}^{5}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b/x^5)^(1/2),x)

[Out]

int(1/x/(a+b/x^5)^(1/2),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/(sqrt(a + b/x^5)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.725371, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-{\left (8 \, a^{2} x^{10} + 8 \, a b x^{5} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{10} + a b x^{5}\right )} \sqrt{\frac{a x^{5} + b}{x^{5}}}\right )}{10 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{5} \sqrt{\frac{a x^{5} + b}{x^{5}}}}{2 \, a x^{5} + b}\right )}{5 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^5)*x),x, algorithm="fricas")

[Out]

[1/10*log(-(8*a^2*x^10 + 8*a*b*x^5 + b^2)*sqrt(a) - 4*(2*a^2*x^10 + a*b*x^5)*sqr
t((a*x^5 + b)/x^5))/sqrt(a), -1/5*sqrt(-a)*arctan(2*sqrt(-a)*x^5*sqrt((a*x^5 + b
)/x^5)/(2*a*x^5 + b))/a]

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Sympy [A]  time = 5.4739, size = 24, normalized size = 0.89 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b/x**5)**(1/2),x)

[Out]

2*asinh(sqrt(a)*x**(5/2)/sqrt(b))/(5*sqrt(a))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{5}}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^5)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^5)*x), x)