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

Optimal. Leaf size=31 \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.120606, 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 \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

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]*x^3)/(2*Sqrt[a - b*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0156379, size = 31, normalized size = 1. \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

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]*x^3)/(2*Sqrt[a - b*x^2])

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Maple [A]  time = 0.003, size = 31, normalized size = 1. \[{\frac{{x}^{3}}{2}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.726032, size = 7, normalized size = 0.23 \[ -\frac{1}{2} i \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*I*x^2

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Fricas [A]  time = 0.274983, size = 55, normalized size = 1.77 \[ -\frac{\sqrt{-b x^{2} + a} x^{3} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{2 \,{\left (b x^{2} - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273877, size = 53, normalized size = 1.71 \[ -\frac{{\left (b x^{2} - a\right )} i{\rm sign}\left (b x^{2} - a\right ){\rm sign}\left (x\right )}{2 \, b} + \frac{a i{\rm sign}\left (a\right ){\rm sign}\left (x\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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