3.581 \(\int \frac{\sqrt{b-\frac{a}{x}} x^m}{\sqrt{a-b x}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}} \]

[Out]

(2*Sqrt[b - a/x]*x^(1 + m))/((1 + 2*m)*Sqrt[a - b*x])

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Rubi [A]  time = 0.036967, antiderivative size = 36, 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} \[ \frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b - a/x]*x^(1 + m))/((1 + 2*m)*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^m}{\sqrt{a-b x}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{x^{-\frac{1}{2}+m} \sqrt{-a+b x}}{\sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int x^{-\frac{1}{2}+m} \, dx}{\sqrt{a-b x}}\\ &=\frac{2 \sqrt{b-\frac{a}{x}} x^{1+m}}{(1+2 m) \sqrt{a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0212461, size = 35, normalized size = 0.97 \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x}}}{\left (m+\frac{1}{2}\right ) \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b - a/x]*x^(1 + m))/((1/2 + m)*Sqrt[a - b*x])

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Maple [A]  time = 0.003, size = 36, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{1+m}}{ \left ( 1+2\,m \right ) \sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.75715, size = 20, normalized size = 0.56 \begin{align*} \frac{2 \, \sqrt{x} x^{m}}{2 i \, m + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*x^m/(2*I*m + I)

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Fricas [A]  time = 1.56237, size = 95, normalized size = 2.64 \begin{align*} \frac{2 \, \sqrt{-b x + a} x x^{m} \sqrt{\frac{b x - a}{x}}}{2 \, a m -{\left (2 \, b m + b\right )} x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2424, size = 212, normalized size = 5.89 \begin{align*} \frac{2 \, \sqrt{-a b} a{\left | b \right |} e^{\left (m \log \left (\frac{a}{b}\right ) - \log \left (\frac{a}{b}\right )\right )} \mathrm{sgn}\left (x\right )}{2 \, b^{3} m + b^{3}} - \frac{2 \,{\left (\frac{\sqrt{-a b} a e^{\left (m \log \left (\frac{a}{b}\right ) - \log \left (\frac{a}{b}\right )\right )}}{2 \, m + 1} + \frac{{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}} e^{\left (m \log \left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right ) - \log \left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right )\right )}}{b{\left (2 \, m + 1\right )}}\right )}{\left | b \right |} \mathrm{sgn}\left (x\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(-a*b)*a*abs(b)*e^(m*log(a/b) - log(a/b))*sgn(x)/(2*b^3*m + b^3) - 2*(sqrt(-a*b)*a*e^(m*log(a/b) - log(a
/b))/(2*m + 1) + (-(b*x - a)*b - a*b)^(3/2)*e^(m*log(((b*x - a)*b + a*b)/b^2) - log(((b*x - a)*b + a*b)/b^2))/
(b*(2*m + 1)))*abs(b)*sgn(x)/b^3