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

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

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Rubi [A]  time = 0.089545, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

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Rubi in Sympy [A]  time = 4.93753, size = 20, normalized size = 0.71 \[ - \frac{\sqrt{a - b x^{2}}}{\sqrt{- \frac{a}{x^{2}} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a - b*x**2)/sqrt(-a/x**2 + b)

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Mathematica [A]  time = 0.0283499, size = 28, normalized size = 1. \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

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Maple [A]  time = 0.015, size = 42, normalized size = 1.5 \[ -{\frac{{x}^{2}}{b{x}^{2}-a}\sqrt{{\frac{b{x}^{2}-a}{{x}^{2}}}}\sqrt{-b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.722078, size = 9, normalized size = 0.32 \[ -i \, \sqrt{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*sqrt(x^2)

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Fricas [A]  time = 0.267732, size = 55, normalized size = 1.96 \[ -\frac{\sqrt{-b x^{2} + a} x^{2} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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