Optimal. Leaf size=43 \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.0329957, antiderivative size = 43, 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 \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/Sqrt[2 + b*x],x]
[Out]
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Rubi in Sympy [A] time = 4.93574, size = 39, normalized size = 0.91 \[ \frac{\sqrt{x} \sqrt{b x + 2}}{b} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(b*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0288893, size = 43, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/Sqrt[2 + b*x],x]
[Out]
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Maple [A] time = 0.006, size = 62, normalized size = 1.4 \[{\frac{1}{b}\sqrt{x}\sqrt{bx+2}}-{1\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(b*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(x)/sqrt(b*x + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218553, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2} \sqrt{b} \sqrt{x} + \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{b^{\frac{3}{2}}}, \frac{\sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{\sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(b*x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.75506, size = 54, normalized size = 1.26 \[ \frac{x^{\frac{3}{2}}}{\sqrt{b x + 2}} + \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(b*x+2)**(1/2),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(x)/sqrt(b*x + 2),x, algorithm="giac")
[Out]