Optimal. Leaf size=45 \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]
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Rubi [A] time = 0.0355879, antiderivative size = 45, 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 \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 5.06434, size = 41, normalized size = 0.91 \[ 2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} - \frac{2 \sqrt{a + b x}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0218209, size = 48, normalized size = 1.07 \[ 2 \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/x^(3/2),x]
[Out]
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Maple [A] time = 0.102, size = 61, normalized size = 1.4 \[ -2\,{\frac{\sqrt{bx+a}}{\sqrt{x}}}+{1\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(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 + a)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253428, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \, \sqrt{b x + a} \sqrt{x}}{x}, \frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) - \sqrt{b x + a} \sqrt{x}\right )}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.86608, size = 68, normalized size = 1.51 \[ - \frac{2 \sqrt{a}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{2 b \sqrt{x}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 12.5019, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/x^(3/2),x, algorithm="giac")
[Out]