Optimal. Leaf size=58 \[ -\frac{2 (b x+2)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{b x+2}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
[Out]
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Rubi [A] time = 0.0408551, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 (b x+2)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{b x+2}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 + b*x)^(3/2)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.71353, size = 56, normalized size = 0.97 \[ 6 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + 3 b \sqrt{x} \sqrt{b x + 2} - \frac{2 \left (b x + 2\right )^{\frac{3}{2}}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+2)**(3/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0481059, size = 45, normalized size = 0.78 \[ \frac{\sqrt{b x+2} (b x-4)}{\sqrt{x}}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(2 + b*x)^(3/2)/x^(3/2),x]
[Out]
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Maple [A] time = 0.025, size = 72, normalized size = 1.2 \[{({b}^{2}{x}^{2}-2\,bx-8){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}}+3\,{\frac{\sqrt{b}\sqrt{x \left ( bx+2 \right ) }}{\sqrt{x}\sqrt{bx+2}}\ln \left ({\frac{bx+1}{\sqrt{b}}}+\sqrt{b{x}^{2}+2\,x} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+2)^(3/2)/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((b*x + 2)^(3/2)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222048, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) + \sqrt{b x + 2}{\left (b x - 4\right )} \sqrt{x}}{x}, \frac{6 \, \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) + \sqrt{b x + 2}{\left (b x - 4\right )} \sqrt{x}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.6304, size = 73, normalized size = 1.26 \[ 6 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{b x + 2}} - \frac{2 b \sqrt{x}}{\sqrt{b x + 2}} - \frac{8}{\sqrt{x} \sqrt{b x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+2)**(3/2)/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((b*x + 2)^(3/2)/x^(3/2),x, algorithm="giac")
[Out]