Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.0117361, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 63, 205} \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{a+b x} \, dx &=\frac{2 \sqrt{x}}{b}-\frac{a \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{b}\\ &=\frac{2 \sqrt{x}}{b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0092629, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{b}}-2\,{\frac{a}{b\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59417, size = 189, normalized size = 4.72 \begin{align*} \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \, \sqrt{x}}{b}, -\frac{2 \,{\left (\sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) - \sqrt{x}\right )}}{b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05438, size = 92, normalized size = 2.3 \begin{align*} \begin{cases} \frac{i \sqrt{a} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} - \frac{i \sqrt{a} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} + \frac{2 \sqrt{x}}{b} & \text{for}\: b \neq 0 \\\frac{2 x^{\frac{3}{2}}}{3 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16379, size = 42, normalized size = 1.05 \begin{align*} -\frac{2 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{2 \, \sqrt{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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