3.1922 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]

[Out]

(Sqrt[a + b/x^2]*x)/a

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Rubi [A]  time = 0.0108583, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

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Rubi in Sympy [A]  time = 1.24666, size = 12, normalized size = 0.75 \[ \frac{x \sqrt{a + \frac{b}{x^{2}}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b/x**2)/a

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Mathematica [A]  time = 0.0112631, size = 16, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

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Maple [A]  time = 0.003, size = 28, normalized size = 1.8 \[{\frac{a{x}^{2}+b}{ax}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(1/2),x)

[Out]

(a*x^2+b)/a/x/((a*x^2+b)/x^2)^(1/2)

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Maxima [A]  time = 1.44192, size = 19, normalized size = 1.19 \[ \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b/x^2)*x/a

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Fricas [A]  time = 0.229905, size = 24, normalized size = 1.5 \[ \frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt((a*x^2 + b)/x^2)/a

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Sympy [A]  time = 2.07586, size = 17, normalized size = 1.06 \[ \frac{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*sqrt(a*x**2/b + 1)/a

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GIAC/XCAS [A]  time = 0.224326, size = 38, normalized size = 2.38 \[ -\frac{\sqrt{b}{\rm sign}\left (x\right )}{a} + \frac{\sqrt{a x^{2} + b}}{a{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(a + b/x^2),x, algorithm="giac")

[Out]

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