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

Optimal. Leaf size=29 \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]

[Out]

2/3*x^2*(b-a/x)^(1/2)/(-b*x+a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {515, 23, 30} \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b - a/x]*x^2)/(3*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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {b-\frac {a}{x}} x}{\sqrt {a-b x}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {\sqrt {x} \sqrt {-a+b x}}{\sqrt {a-b x}} \, dx}{\sqrt {-a+b x}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \sqrt {x} \, dx}{\sqrt {a-b x}}\\ &=\frac {2 \sqrt {b-\frac {a}{x}} x^2}{3 \sqrt {a-b x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b - a/x]*x^2)/(3*Sqrt[a - b*x])

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fricas [A]  time = 0.40, size = 35, normalized size = 1.21 \[ -\frac {2 \, \sqrt {-b x + a} x^{2} \sqrt {\frac {b x - a}{x}}}{3 \, {\left (b x - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.49, size = 56, normalized size = 1.93 \[ \frac {2 \, \sqrt {-a b} a {\left | b \right |} \mathrm {sgn}\relax (x)}{3 \, b^{3}} - \frac {2 \, {\left (\sqrt {-a b} a + \frac {{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac {3}{2}}}{b}\right )} {\left | b \right |} \mathrm {sgn}\relax (x)}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.93 \[ \frac {2 \sqrt {-\frac {-b x +a}{x}}\, x^{2}}{3 \sqrt {-b x +a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^2*(-(-b*x+a)/x)^(1/2)/(-b*x+a)^(1/2)

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maxima [C]  time = 0.73, size = 5, normalized size = 0.17 \[ -\frac {2}{3} i \, x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.06, size = 23, normalized size = 0.79 \[ \frac {2\,x^2\,\sqrt {b-\frac {a}{x}}}{3\,\sqrt {a-b\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^2*(b - a/x)^(1/2))/(3*(a - b*x)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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