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

Optimal. Leaf size=24 \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

[Out]

(-2*Sqrt[b - a/x])/Sqrt[a - b*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.129611, antiderivative size = 24, 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 \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b - a/x])/Sqrt[a - b*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.55558, size = 19, normalized size = 0.79 \[ \frac{2 \sqrt{a - b x}}{x \sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a - b*x)/(x*sqrt(-a/x + b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0265813, size = 24, normalized size = 1. \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b - a/x])/Sqrt[a - b*x]

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 24, normalized size = 1. \[ -2\,{\frac{1}{\sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.757043, size = 7, normalized size = 0.29 \[ \frac{2 i}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I/sqrt(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.262101, size = 43, normalized size = 1.79 \[ \frac{2 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}}{b x - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287364, size = 57, normalized size = 2.38 \[ \frac{2 \,{\left (\frac{b^{3}}{\sqrt{-{\left (b x - a\right )} b - a b}} - \frac{b^{3}}{\sqrt{-a b}}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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