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

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 = 0.130518, 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 \[ \frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}} \]

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])

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Rubi in Sympy [A]  time = 5.74633, size = 36, normalized size = 1. \[ - \frac{2 x^{m + \frac{1}{2}} \sqrt{a - b x}}{\sqrt{x} \left (2 m + 1\right ) \sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b-a/x)**(1/2)/(-b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.133393, size = 34, normalized size = 0.94 \[ -\frac{2 x^m \sqrt{a-b x}}{(2 m+1) \sqrt{b-\frac{a}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 1. \[ 2\,{\frac{{x}^{1+m}}{ \left ( 1+2\,m \right ) \sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \]

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 [A]  time = 0.740365, size = 20, normalized size = 0.56 \[ \frac{2 \, \sqrt{x} x^{m}}{2 i \, m + i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x)*x^m/sqrt(-b*x + a),x, algorithm="maxima")

[Out]

2*sqrt(x)*x^m/(2*I*m + I)

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Fricas [A]  time = 0.288123, size = 59, normalized size = 1.64 \[ \frac{2 \, \sqrt{-b x + a} x x^{m} \sqrt{\frac{b x - a}{x}}}{2 \, a m -{\left (2 \, b m + b\right )} x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x)*x^m/sqrt(-b*x + a),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. \[ \int \frac{x^{m} \sqrt{- \frac{a}{x} + b}}{\sqrt{a - b x}}\, dx \]

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/XCAS [A]  time = 0.29791, size = 212, normalized size = 5.89 \[ \frac{2 \, \sqrt{-a b} a{\left | b \right |} e^{\left (m{\rm ln}\left (\frac{a}{b}\right ) -{\rm ln}\left (\frac{a}{b}\right )\right )}{\rm sign}\left (x\right )}{2 \, b^{3} m + b^{3}} - \frac{2 \,{\left (\frac{\sqrt{-a b} a e^{\left (m{\rm ln}\left (\frac{a}{b}\right ) -{\rm ln}\left (\frac{a}{b}\right )\right )}}{2 \, m + 1} + \frac{{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}} e^{\left (m{\rm ln}\left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right ) -{\rm ln}\left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right )\right )}}{b{\left (2 \, m + 1\right )}}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x)*x^m/sqrt(-b*x + a),x, algorithm="giac")

[Out]

2*sqrt(-a*b)*a*abs(b)*e^(m*ln(a/b) - ln(a/b))*sign(x)/(2*b^3*m + b^3) - 2*(sqrt(
-a*b)*a*e^(m*ln(a/b) - ln(a/b))/(2*m + 1) + (-(b*x - a)*b - a*b)^(3/2)*e^(m*ln((
(b*x - a)*b + a*b)/b^2) - ln(((b*x - a)*b + a*b)/b^2))/(b*(2*m + 1)))*abs(b)*sig
n(x)/b^3