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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {2 x \sqrt {b-\frac {a}{x}}}{\sqrt {a-b x}} \]

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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fricas [A]  time = 0.40, size = 33, normalized size = 1.32 \[ -\frac {2 \, \sqrt {-b x + a} x \sqrt {\frac {b x - a}{x}}}{b x - a} \]

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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giac [B]  time = 0.40, size = 51, normalized size = 2.04 \[ \frac {2 \, {\left (\sqrt {-{\left (b x - a\right )} b - a b} - \sqrt {-a b}\right )} {\left | b \right |} \mathrm {sgn}\relax (x)}{b^{2}} + \frac {2 \, \sqrt {-a b} {\left | b \right |} \mathrm {sgn}\relax (x)}{b^{2}} \]

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

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maple [A]  time = 0.00, size = 25, normalized size = 1.00 \[ \frac {2 \sqrt {-\frac {-b x +a}{x}}\, x}{\sqrt {-b x +a}} \]

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 = 0.67, size = 5, normalized size = 0.20 \[ -2 i \, \sqrt {x} \]

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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mupad [B]  time = 3.04, size = 21, normalized size = 0.84 \[ \frac {2\,x\,\sqrt {b-\frac {a}{x}}}{\sqrt {a-b\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b - a/x)^(1/2)/(a - b*x)^(1/2),x)

[Out]

(2*x*(b - a/x)^(1/2))/(a - b*x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \]

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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