Optimal. Leaf size=41 \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]
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Rubi [A] time = 0.0322514, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + b*x]/x^(3/2),x]
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Rubi in Sympy [A] time = 4.85653, size = 39, normalized size = 0.95 \[ 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{2 \sqrt{b x + 2}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+2)**(1/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0228564, size = 41, normalized size = 1. \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + b*x]/x^(3/2),x]
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Maple [A] time = 0.033, size = 59, normalized size = 1.4 \[ -2\,{\frac{\sqrt{bx+2}}{\sqrt{x}}}+{1\sqrt{b}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+2)^(1/2)/x^(3/2),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 + 2)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21834, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) - 2 \, \sqrt{b x + 2} \sqrt{x}}{x}, \frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) - \sqrt{b x + 2} \sqrt{x}\right )}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + 2)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.46726, size = 48, normalized size = 1.17 \[ - 2 \sqrt{b} \sqrt{1 + \frac{2}{b x}} - \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 \sqrt{b} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+2)**(1/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + 2)/x^(3/2),x, algorithm="giac")
[Out]