Optimal. Leaf size=72 \[ \frac{2 B \sqrt{b x+c x^2}}{c \sqrt{e x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{b x+c x^2}}{\sqrt{b} \sqrt{e x}}\right )}{\sqrt{b} \sqrt{e}} \]
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Rubi [A] time = 0.0611555, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {794, 660, 208} \[ \frac{2 B \sqrt{b x+c x^2}}{c \sqrt{e x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{b x+c x^2}}{\sqrt{b} \sqrt{e x}}\right )}{\sqrt{b} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 660
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{e x} \sqrt{b x+c x^2}} \, dx &=\frac{2 B \sqrt{b x+c x^2}}{c \sqrt{e x}}+A \int \frac{1}{\sqrt{e x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{2 B \sqrt{b x+c x^2}}{c \sqrt{e x}}+(2 A e) \operatorname{Subst}\left (\int \frac{1}{-b e+e^2 x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{e x}}\right )\\ &=\frac{2 B \sqrt{b x+c x^2}}{c \sqrt{e x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{b x+c x^2}}{\sqrt{b} \sqrt{e x}}\right )}{\sqrt{b} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0354381, size = 71, normalized size = 0.99 \[ \frac{2 x \left (\sqrt{b} B (b+c x)-A c \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{\sqrt{b} c \sqrt{e x} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 72, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{\sqrt{ex}\sqrt{e \left ( cx+b \right ) }c\sqrt{be}} \left ( Ace{\it Artanh} \left ({\frac{\sqrt{e \left ( cx+b \right ) }}{\sqrt{be}}} \right ) -B\sqrt{e \left ( cx+b \right ) }\sqrt{be} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65939, size = 367, normalized size = 5.1 \begin{align*} \left [\frac{\sqrt{b e} A c x \log \left (-\frac{c e x^{2} + 2 \, b e x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b e} \sqrt{e x}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{e x} B b}{b c e x}, \frac{2 \,{\left (\sqrt{-b e} A c x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-b e} \sqrt{e x}}{c e x^{2} + b e x}\right ) + \sqrt{c x^{2} + b x} \sqrt{e x} B b\right )}}{b c e 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}{\sqrt{e 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 = 1.15076, size = 139, normalized size = 1.93 \begin{align*} 2 \,{\left (\frac{A \arctan \left (\frac{\sqrt{c x e + b e}}{\sqrt{-b e}}\right ) e}{\sqrt{-b e}} + \frac{\sqrt{c x e + b e} B}{c}\right )} e^{\left (-1\right )} - \frac{2 \,{\left (A c \arctan \left (\frac{\sqrt{b} e^{\frac{1}{2}}}{\sqrt{-b e}}\right ) e + \sqrt{-b e} B \sqrt{b} e^{\frac{1}{2}}\right )} e^{\left (-1\right )}}{\sqrt{-b e} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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