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

Optimal. Leaf size=26 \[ -\frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

[Out]

-(Sqrt[b - a/x^2]/Sqrt[a - b*x^2])

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Rubi [A]  time = 0.112253, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[b - a/x^2]/Sqrt[a - b*x^2])

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Rubi in Sympy [A]  time = 5.9837, size = 22, normalized size = 0.85 \[ \frac{\sqrt{a - b x^{2}}}{x^{2} \sqrt{- \frac{a}{x^{2}} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0334872, size = 26, normalized size = 1. \[ -\frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[b - a/x^2]/Sqrt[a - b*x^2])

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Maple [A]  time = 0.004, size = 28, normalized size = 1.1 \[ -{1\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.731808, size = 9, normalized size = 0.35 \[ \frac{i}{\sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I/sqrt(x^2)

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Fricas [A]  time = 0.266722, size = 55, normalized size = 2.12 \[ -\frac{\sqrt{-b x^{2} + a}{\left (x - 1\right )} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b - \frac{a}{x^{2}}}}{\sqrt{-b x^{2} + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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