3.423 \(\int \frac{\sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

[Out]

(2*Sqrt[b - a/x]*x)/Sqrt[a - b*x]

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Rubi [A]  time = 0.042153, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b - a/x]*x)/Sqrt[a - b*x]

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Rubi in Sympy [A]  time = 4.35024, size = 19, normalized size = 0.76 \[ - \frac{2 \sqrt{a - b x}}{\sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a - b*x)/sqrt(-a/x + b)

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Mathematica [A]  time = 0.0265951, size = 25, normalized size = 1. \[ \frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b - a/x]*x)/Sqrt[a - b*x]

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \[ 2\,{\frac{x}{\sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.783961, size = 7, normalized size = 0.28 \[ -2 i \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*I*sqrt(x)

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Fricas [A]  time = 0.260108, size = 41, normalized size = 1.64 \[ \frac{2 \,{\left (b x - a\right )}}{\sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282074, size = 69, normalized size = 2.76 \[ \frac{2 \,{\left (\sqrt{-{\left (b x - a\right )} b - a b} - \sqrt{-a b}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{b^{2}} + \frac{2 \, \sqrt{-a b}{\left | b \right |}{\rm sign}\left (x\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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