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

Optimal. Leaf size=29 \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}} \]

[Out]

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

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Rubi [A]  time = 0.123482, antiderivative size = 29, 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 \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.40434, size = 22, normalized size = 0.76 \[ \frac{2 \sqrt{a - b x}}{3 x^{2} \sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0339624, size = 29, normalized size = 1. \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.9 \[ -{\frac{2}{3\,x}\sqrt{-{\frac{-bx+a}{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.802035, size = 7, normalized size = 0.24 \[ \frac{2 i}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*I/x^(3/2)

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Fricas [A]  time = 0.265394, size = 47, normalized size = 1.62 \[ \frac{2 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}}{3 \,{\left (b x^{2} - a x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282479, size = 81, normalized size = 2.79 \[ \frac{2 \,{\left (\frac{b^{5}}{{\left ({\left (b x - a\right )} b + a b\right )} \sqrt{-{\left (b x - a\right )} b - a b}} - \frac{b^{4}}{\sqrt{-a b} a}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^5/(((b*x - a)*b + a*b)*sqrt(-(b*x - a)*b - a*b)) - b^4/(sqrt(-a*b)*a))*ab
s(b)*sign(x)/b^3