Optimal. Leaf size=44 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]
[Out]
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Rubi [A] time = 0.0325928, antiderivative size = 44, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(2 + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 5.46941, size = 41, normalized size = 0.93 \[ - \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} + \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(b*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0324645, size = 44, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(2 + b*x)^(3/2),x]
[Out]
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Maple [A] time = 0.114, size = 48, normalized size = 1.1 \[ 2\,{\frac{1}{{b}^{3/2}\sqrt{\pi }} \left ( -1/2\,{\frac{\sqrt{\pi }\sqrt{x}\sqrt{2}\sqrt{b}}{\sqrt{1/2\,bx+1}}}+\sqrt{\pi }{\it Arcsinh} \left ( 1/2\,\sqrt{b}\sqrt{x}\sqrt{2} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(b*x+2)^(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(x)/(b*x + 2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220105, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2} \sqrt{x} \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right ) - 2 \, \sqrt{b} x}{\sqrt{b x + 2} b^{\frac{3}{2}} \sqrt{x}}, \frac{2 \,{\left (\sqrt{b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) - \sqrt{-b} x\right )}}{\sqrt{b x + 2} \sqrt{-b} b \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(b*x + 2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.95379, size = 41, normalized size = 0.93 \[ - \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} + \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(b*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222615, size = 111, normalized size = 2.52 \[ -\frac{{\left (\frac{{\rm ln}\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{\sqrt{b}} + \frac{8 \, \sqrt{b}}{{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b}\right )}{\left | b \right |}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(b*x + 2)^(3/2),x, algorithm="giac")
[Out]