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

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 = 0.0330215, 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, Rules used = {515, 23, 30} \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}} \]

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])

Rule 515

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Dist[(x^(n*FracPa
rt[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x], x] /
; FreeQ[{a, b, c, d, m, 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}}}{x^2 \sqrt{a-b x}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{\sqrt{-a+b x}}{x^{5/2} \sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{1}{x^{5/2}} \, dx}{\sqrt{a-b x}}\\ &=-\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0107928, 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 = 0.002, size = 27, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,x}\sqrt{-{\frac{-bx+a}{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}

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 [C]  time = 1.61323, size = 7, normalized size = 0.24 \begin{align*} \frac{2 i}{3 \, x^{\frac{3}{2}}} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.40859, size = 70, normalized size = 2.41 \begin{align*} \frac{2 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}}{3 \,{\left (b x^{2} - a x\right )}} \end{align*}

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, 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. \begin{align*} \int \frac{\sqrt{- \frac{a}{x} + b}}{x^{2} \sqrt{a - b x}}\, dx \end{align*}

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 [B]  time = 1.14102, size = 81, normalized size = 2.79 \begin{align*} \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 |} \mathrm{sgn}\left (x\right )}{3 \, b^{3}} \end{align*}

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, 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))*abs(b)*sgn(x)/b^3