Optimal. Leaf size=64 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
[Out]
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Rubi [A] time = 0.0450904, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*Sqrt[2 + b*x],x]
[Out]
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Rubi in Sympy [A] time = 6.4283, size = 56, normalized size = 0.88 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{2 b} - \frac{\sqrt{x} \sqrt{b x + 2}}{2 b} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
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 = 0.0440559, size = 51, normalized size = 0.8 \[ \frac{\sqrt{x} (b x+1) \sqrt{b x+2}}{2 b}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*Sqrt[2 + b*x],x]
[Out]
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Maple [A] time = 0.02, size = 75, normalized size = 1.2 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{1}{2\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{1}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]
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(b*x + 2)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223674, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2}{\left (b x + 1\right )} \sqrt{b} \sqrt{x} + \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{2 \, b^{\frac{3}{2}}}, \frac{\sqrt{b x + 2}{\left (b x + 1\right )} \sqrt{-b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + 2)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.69718, size = 71, normalized size = 1.11 \[ \frac{b x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{3 x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{\sqrt{x}}{b \sqrt{b x + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
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(b*x + 2)*sqrt(x),x, algorithm="giac")
[Out]