Optimal. Leaf size=27 \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]
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Rubi [A] time = 0.0393953, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[x])^(-1),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \log{\left (a + b \sqrt{x} \right )}}{b^{2}} + 2 \int ^{\sqrt{x}} \frac{1}{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**(1/2)),x)
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Mathematica [A] time = 0.0108455, size = 27, normalized size = 1. \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[x])^(-1),x]
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Maple [B] time = 0.013, size = 57, normalized size = 2.1 \[ 2\,{\frac{\sqrt{x}}{b}}+{\frac{a}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) }-{\frac{a}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) }-{\frac{a\ln \left ({b}^{2}x-{a}^{2} \right ) }{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^(1/2)),x)
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Maxima [A] time = 1.44789, size = 36, normalized size = 1.33 \[ -\frac{2 \, a \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*sqrt(x) + a),x, algorithm="maxima")
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Fricas [A] time = 0.234246, size = 30, normalized size = 1.11 \[ -\frac{2 \,{\left (a \log \left (b \sqrt{x} + a\right ) - b \sqrt{x}\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*sqrt(x) + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.424704, size = 27, normalized size = 1. \[ \begin{cases} - \frac{2 a \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{2}} + \frac{2 \sqrt{x}}{b} & \text{for}\: b \neq 0 \\\frac{x}{a} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*x**(1/2)),x)
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GIAC/XCAS [A] time = 0.215655, size = 32, normalized size = 1.19 \[ -\frac{2 \, a{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, \sqrt{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*sqrt(x) + a),x, algorithm="giac")
[Out]