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

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 = 0.014152, 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, Rules used = {435, 23, 30} \[ \frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]

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]

Rule 435

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(x^(n*FracPart[q])*(c +
d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b,
c, d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{\sqrt{-a+b x}}{\sqrt{x} \sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x}} \, dx}{\sqrt{a-b x}}\\ &=\frac{2 \sqrt{b-\frac{a}{x}} x}{\sqrt{a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.010256, 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. \begin{align*} 2\,{\frac{x}{\sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \end{align*}

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 [C]  time = 1.38986, size = 7, normalized size = 0.28 \begin{align*} -2 i \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b-a/x)^(1/2)/(-b*x+a)^(1/2),x, algorithm="maxima")

[Out]

-2*I*sqrt(x)

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Fricas [A]  time = 1.46328, size = 66, normalized size = 2.64 \begin{align*} -\frac{2 \, \sqrt{-b x + a} x \sqrt{\frac{b x - a}{x}}}{b x - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b-a/x)^(1/2)/(-b*x+a)^(1/2),x, algorithm="fricas")

[Out]

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

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

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 [B]  time = 1.11791, size = 69, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (\sqrt{-{\left (b x - a\right )} b - a b} - \sqrt{-a b}\right )}{\left | b \right |} \mathrm{sgn}\left (x\right )}{b^{2}} + \frac{2 \, \sqrt{-a b}{\left | b \right |} \mathrm{sgn}\left (x\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b-a/x)^(1/2)/(-b*x+a)^(1/2),x, algorithm="giac")

[Out]

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