Optimal. Leaf size=39 \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0350426, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/x^2,x]
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Rubi in Sympy [A] time = 2.39471, size = 32, normalized size = 0.82 \[ - \frac{\sqrt{a + b x}}{x} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**2,x)
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Mathematica [A] time = 0.0326559, size = 39, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/x^2,x]
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Maple [A] time = 0.007, size = 37, normalized size = 1. \[ 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/x^2,x)
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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 = 0.229515, size = 1, normalized size = 0.03 \[ \left [\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \, \sqrt{b x + a} \sqrt{a}}{2 \, \sqrt{a} x}, \frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} \sqrt{-a}}{\sqrt{-a} x}\right ] \]
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 [A] time = 2.92092, size = 44, normalized size = 1.13 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/x**2,x)
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GIAC/XCAS [A] time = 0.205324, size = 55, normalized size = 1.41 \[ \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x + a} b}{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/x^2,x, algorithm="giac")
[Out]