Optimal. Leaf size=29 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.013797, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (b+a x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0074179, size = 29, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 19, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{\sqrt{ab}}\arctan \left ({\frac{a\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 = 2.01815, size = 163, normalized size = 5.62 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right )}{a b}, -\frac{2 \, \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right )}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58298, size = 94, normalized size = 3.24 \begin{align*} \begin{cases} \tilde{\infty } \sqrt{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\\frac{2 \sqrt{x}}{b} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} + \frac{i \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05216, size = 24, normalized size = 0.83 \begin{align*} \frac{2 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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