Optimal. Leaf size=51 \[ 2 x \sqrt{\frac{a}{x^2}+\frac{b}{x}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+\frac{b}{x}}}\right ) \]
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Rubi [A] time = 0.125978, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ 2 x \sqrt{\frac{a}{x^2}+\frac{b}{x}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+\frac{b}{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(a + b*x)/x^2],x]
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Rubi in Sympy [A] time = 8.51486, size = 41, normalized size = 0.8 \[ - 2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + \frac{b}{x}}} \right )} + 2 x \sqrt{\frac{a}{x^{2}} + \frac{b}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0456024, size = 58, normalized size = 1.14 \[ \frac{2 x \sqrt{\frac{a+b x}{x^2}} \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(a + b*x)/x^2],x]
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Maple [A] time = 0.01, size = 47, normalized size = 0.9 \[ 2\,{\frac{x}{\sqrt{bx+a}}\sqrt{{\frac{bx+a}{{x}^{2}}}} \left ( -\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +\sqrt{bx+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)/x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)/x^2),x, algorithm="maxima")
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Fricas [A] time = 0.23282, size = 1, normalized size = 0.02 \[ \left [2 \, x \sqrt{\frac{b x + a}{x^{2}}} + \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{a} x \sqrt{\frac{b x + a}{x^{2}}} + 2 \, a}{x}\right ), 2 \, x \sqrt{\frac{b x + a}{x^{2}}} - 2 \, \sqrt{-a} \arctan \left (\frac{x \sqrt{\frac{b x + a}{x^{2}}}}{\sqrt{-a}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)/x^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{a + b x}{x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)/x**2)**(1/2),x)
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GIAC/XCAS [A] time = 0.222044, size = 88, normalized size = 1.73 \[ 2 \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x + a}\right )}{\rm sign}\left (x\right ) - \frac{2 \,{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )}{\rm sign}\left (x\right )}{\sqrt{-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)/x^2),x, algorithm="giac")
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