Optimal. Leaf size=40 \[ \frac{2 B \sqrt{a+b x}}{b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0115786, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 63, 208} \[ \frac{2 B \sqrt{a+b x}}{b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x \sqrt{a+b x}} \, dx &=\frac{2 B \sqrt{a+b x}}{b}+A \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=\frac{2 B \sqrt{a+b x}}{b}+\frac{(2 A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=\frac{2 B \sqrt{a+b x}}{b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0165682, size = 40, normalized size = 1. \[ \frac{2 B \sqrt{a+b x}}{b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{b} \left ( B\sqrt{bx+a}-{\frac{Ab}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \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 = 2.78031, size = 228, normalized size = 5.7 \begin{align*} \left [\frac{A \sqrt{a} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} B a}{a b}, \frac{2 \,{\left (A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + \sqrt{b x + a} B a\right )}}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.13169, size = 56, normalized size = 1.4 \begin{align*} \frac{2 A \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x}} \right )}}{a \sqrt{- \frac{1}{a}}} - B \left (\begin{cases} - \frac{x}{\sqrt{a}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + b x}}{b} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15543, size = 49, normalized size = 1.22 \begin{align*} \frac{2 \, A \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \, \sqrt{b x + a} B}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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