3.6.84 \(\int \frac {\sqrt {b-\frac {a}{x}}}{\sqrt {a-b x}} \, dx\) [584]

Optimal. Leaf size=25 \[ \frac {2 \sqrt {b-\frac {a}{x}} x}{\sqrt {a-b x}} \]

[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 = {446, 23, 30} \begin {gather*} \frac {2 x \sqrt {b-\frac {a}{x}}}{\sqrt {a-b x}} \end {gather*}

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 446

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 = 4.78, size = 24, normalized size = 0.96 \begin {gather*} -\frac {2 \sqrt {a-b x}}{\sqrt {b-\frac {a}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b - a/x]/Sqrt[a - b*x],x]

[Out]

(-2*Sqrt[a - b*x])/Sqrt[b - a/x]

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Maple [A]
time = 0.29, size = 25, normalized size = 1.00

method	result	size
gosper	\(\frac {2 x \sqrt {-\frac {-b x +a}{x}}}{\sqrt {-b x +a}}\)	\(25\)
default	\(\frac {2 x \sqrt {-\frac {-b x +a}{x}}}{\sqrt {-b x +a}}\)	\(25\)
risch	\(-\frac {2 \sqrt {-\frac {-b x +a}{x}}\, \sqrt {-\left (-b x +a \right ) x}\, \sqrt {\frac {x \left (-b x +a \right )}{b x -a}}\, \left (b x -a \right ) x}{\left (-b x +a \right )^{\frac {3}{2}} \sqrt {\left (b x -a \right ) x}\, \sqrt {-x}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b-a/x)^(1/2)/(-b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 5, normalized size = 0.20 \begin {gather*} -2 i \, \sqrt {x} \end {gather*}

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 = 0.33, size = 33, normalized size = 1.32 \begin {gather*} -\frac {2 \, \sqrt {-b x + a} x \sqrt {\frac {b x - a}{x}}}{b x - a} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
time = 4.30, size = 51, normalized size = 2.04 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 3.04, size = 21, normalized size = 0.84 \begin {gather*} \frac {2\,x\,\sqrt {b-\frac {a}{x}}}{\sqrt {a-b\,x}} \end {gather*}

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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