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 = 0.0135099, 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, Rules used = {190, 43} \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0120507, 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.
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Maple [B] time = 0.006, size = 57, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{x}}{b}}-{\frac{a}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) }+{\frac{a}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) }-{\frac{a\ln \left ({b}^{2}x-{a}^{2} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968798, size = 36, normalized size = 1.33 \begin{align*} -\frac{2 \, a \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31158, size = 58, normalized size = 2.15 \begin{align*} -\frac{2 \,{\left (a \log \left (b \sqrt{x} + a\right ) - b \sqrt{x}\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.154442, size = 27, normalized size = 1. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10129, size = 32, normalized size = 1.19 \begin{align*} -\frac{2 \, a \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, \sqrt{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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