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

Optimal. Leaf size=31 \[ -\frac{\sqrt{b-\frac{a}{x^2}}}{2 x \sqrt{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.114928, antiderivative size = 31, 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 \[ -\frac{\sqrt{b-\frac{a}{x^2}}}{2 x \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.78084, size = 24, normalized size = 0.77 \[ \frac{\sqrt{a - b x^{2}}}{2 x^{3} \sqrt{- \frac{a}{x^{2}} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0250569, size = 38, normalized size = 1.23 \[ \frac{\sqrt{b-\frac{a}{x^2}} \sqrt{a-b x^2}}{2 b x^3-2 a x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b - a/x^2]*Sqrt[a - b*x^2])/(-2*a*x + 2*b*x^3)

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Maple [A]  time = 0.003, size = 31, normalized size = 1. \[ -{\frac{1}{2\,x}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.774443, size = 7, normalized size = 0.23 \[ \frac{i}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*I/x^2

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Fricas [A]  time = 0.270704, size = 59, normalized size = 1.9 \[ -\frac{\sqrt{-b x^{2} + a}{\left (x^{2} - 1\right )} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{2 \,{\left (b x^{3} - a x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-b*x^2 + a)*(x^2 - 1)*sqrt((b*x^2 - a)/x^2)/(b*x^3 - a*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275152, size = 27, normalized size = 0.87 \[ -\frac{{\rm sign}\left (b x^{2} - a\right ){\rm sign}\left (x\right )}{2 \, i x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sign(b*x^2 - a)*sign(x)/(i*x^2)