Optimal. Leaf size=49 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{2 B \sqrt{x}}{b} \]
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Rubi [A] time = 0.0216724, antiderivative size = 49, 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, 205} \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{2 B \sqrt{x}}{b} \]
Antiderivative was successfully verified.
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Rule 80
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} (a+b x)} \, dx &=\frac{2 B \sqrt{x}}{b}+\frac{\left (2 \left (\frac{A b}{2}-\frac{a B}{2}\right )\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{b}\\ &=\frac{2 B \sqrt{x}}{b}+\frac{\left (4 \left (\frac{A b}{2}-\frac{a B}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 B \sqrt{x}}{b}+\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0267192, size = 49, normalized size = 1. \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{2 B \sqrt{x}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 53, normalized size = 1.1 \begin{align*} 2\,{\frac{B\sqrt{x}}{b}}+2\,{\frac{A}{\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{Ba}{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 = 2.43824, size = 250, normalized size = 5.1 \begin{align*} \left [\frac{2 \, B a b \sqrt{x} +{\left (B a - A b\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right )}{a b^{2}}, \frac{2 \,{\left (B a b \sqrt{x} +{\left (B a - A b\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right )\right )}}{a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38755, size = 218, normalized size = 4.45 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{\sqrt{x}} + 2 B \sqrt{x}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{- \frac{2 A}{\sqrt{x}} + 2 B \sqrt{x}}{b} & \text{for}\: a = 0 \\\frac{2 A \sqrt{x} + \frac{2 B x^{\frac{3}{2}}}{3}}{a} & \text{for}\: b = 0 \\- \frac{i A \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i A \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i B \sqrt{a} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} - \frac{i B \sqrt{a} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} + \frac{2 B \sqrt{x}}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14286, size = 53, normalized size = 1.08 \begin{align*} \frac{2 \, B \sqrt{x}}{b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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