3.437 \(\int \frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

[Out]

(Sqrt[b - a/x^2]*x*Log[x])/Sqrt[a - b*x^2]

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Rubi [A]  time = 0.0530788, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[b - a/x^2]*x*Log[x])/Sqrt[a - b*x^2]

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Rubi in Sympy [A]  time = 5.15388, size = 26, normalized size = 0.93 \[ - \frac{\sqrt{a - b x^{2}} \log{\left (x \right )}}{x \sqrt{- \frac{a}{x^{2}} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a - b*x**2)*log(x)/(x*sqrt(-a/x**2 + b))

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Mathematica [A]  time = 0.11024, size = 0, normalized size = 0. \[ \int \frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

Integrate[Sqrt[b - a/x^2]/Sqrt[a - b*x^2], x]

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Maple [A]  time = 0.011, size = 42, normalized size = 1.5 \[ -{\frac{x\ln \left ( x \right ) }{b{x}^{2}-a}\sqrt{{\frac{b{x}^{2}-a}{{x}^{2}}}}\sqrt{-b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.719973, size = 5, normalized size = 0.18 \[ -i \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*log(x)

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Fricas [A]  time = 0.291698, size = 69, normalized size = 2.46 \[ -\arctan \left (\frac{\sqrt{-b x^{2} + a}{\left (x^{3} + x\right )} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{4} -{\left (a + b\right )} x^{2} + a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(sqrt(-b*x^2 + a)*(x^3 + x)*sqrt((b*x^2 - a)/x^2)/(b*x^4 - (a + b)*x^2 +
a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \frac{a}{x^{2}} + b}}{\sqrt{a - b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-a/x**2 + b)/sqrt(a - b*x**2), x)

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GIAC/XCAS [A]  time = 0.275199, size = 42, normalized size = 1.5 \[ -\frac{1}{2} \, i{\rm ln}\left ({\left (b x^{2} - a\right )} i + a i\right ){\rm sign}\left (b x^{2} - a\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*i*ln((b*x^2 - a)*i + a*i)*sign(b*x^2 - a)*sign(x)