Optimal. Leaf size=52 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 A \sqrt{b x+c x^2}}{b x} \]
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Rubi [A] time = 0.0348563, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 620, 206} \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 A \sqrt{b x+c x^2}}{b x} \]
Antiderivative was successfully verified.
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Rule 792
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{x \sqrt{b x+c x^2}} \, dx &=-\frac{2 A \sqrt{b x+c x^2}}{b x}+B \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 A \sqrt{b x+c x^2}}{b x}+(2 B) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=-\frac{2 A \sqrt{b x+c x^2}}{b x}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.155909, size = 69, normalized size = 1.33 \[ \frac{2 \sqrt{x (b+c x)} \left (\frac{\sqrt{b} B \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{c} \sqrt{\frac{c x}{b}+1}}-A\right )}{b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 51, normalized size = 1. \begin{align*}{B\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}-2\,{\frac{A\sqrt{c{x}^{2}+bx}}{bx}} \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 = 1.94834, size = 262, normalized size = 5.04 \begin{align*} \left [\frac{B b \sqrt{c} x \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + b x} A c}{b c x}, -\frac{2 \,{\left (B b \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x} A c\right )}}{b c x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15437, size = 80, normalized size = 1.54 \begin{align*} -\frac{B \log \left ({\left | 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b \right |}\right )}{\sqrt{c}} + \frac{2 \, A}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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