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