3.2106 \(\int \frac{1}{\sqrt{-a+\frac{b}{x^5}} x} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0666003, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-a + b/x^5]*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.58164, size = 26, normalized size = 0.9 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + \frac{b}{x^{5}}}}{\sqrt{a}} \right )}}{5 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atan(sqrt(-a + b/x**5)/sqrt(a))/(5*sqrt(a))

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Mathematica [A]  time = 0.0658279, 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 = 0.052, 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 = 0.736699, size = 1, normalized size = 0.03 \[ \left [-\frac{\sqrt{-a} \log \left (-{\left (8 \, a^{2} x^{10} - 8 \, a b x^{5} + b^{2}\right )} \sqrt{-a} - 4 \,{\left (2 \, a^{2} x^{10} - a b x^{5}\right )} \sqrt{-\frac{a x^{5} - b}{x^{5}}}\right )}{10 \, a}, -\frac{\arctan \left (\frac{2 \, \sqrt{a} x^{5} \sqrt{-\frac{a x^{5} - b}{x^{5}}}}{2 \, a x^{5} - b}\right )}{5 \, \sqrt{a}}\right ] \]

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*sqrt(-a)*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)*sqrt(-(a*x^5 - b)/x^5))/a, -1/5*arctan(2*sqrt(a)*x^5*sqrt(-(a*x^5 - b)/x
^5)/(2*a*x^5 - b))/sqrt(a)]

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Sympy [A]  time = 5.71677, size = 60, normalized size = 2.07 \[ \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{for}\: \left |{\frac{a x^{5}}{b}}\right | > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*I*acosh(sqrt(a)*x**(5/2)/sqrt(b))/(5*sqrt(a)), Abs(a*x**5/b) > 1),
(2*asin(sqrt(a)*x**(5/2)/sqrt(b))/(5*sqrt(a)), True))

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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)