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

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

[Out]

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

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Rubi [A]  time = 0.0919785, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.68248, size = 22, normalized size = 0.76 \[ - \frac{2 x \sqrt{a - b x}}{3 \sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x*sqrt(a - b*x)/(3*sqrt(-a/x + b))

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Mathematica [A]  time = 0.0291623, size = 29, normalized size = 1. \[ \frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.759298, size = 7, normalized size = 0.24 \[ -\frac{2}{3} i \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.267406, size = 45, normalized size = 1.55 \[ \frac{2 \,{\left (b x^{2} - a x\right )}}{3 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282147, size = 76, normalized size = 2.62 \[ \frac{2 \, \sqrt{-a b} a{\left | b \right |}{\rm sign}\left (x\right )}{3 \, b^{3}} - \frac{2 \,{\left (\sqrt{-a b} a + \frac{{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}}}{b}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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