Optimal. Leaf size=44 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0366256, 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 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 4.88176, size = 39, normalized size = 0.89 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} + \sqrt{x} \sqrt{a + b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0210274, size = 47, normalized size = 1.07 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/Sqrt[x],x]
[Out]
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Maple [A] time = 0.01, size = 62, normalized size = 1.4 \[ \sqrt{x}\sqrt{bx+a}+{\frac{a}{2}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/x^(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 + a)/sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219783, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x}}{2 \, \sqrt{b}}, \frac{a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) + \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{\sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.49882, size = 42, normalized size = 0.95 \[ \sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 12.3072, 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)/sqrt(x),x, algorithm="giac")
[Out]