Optimal. Leaf size=32 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0124113, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {660, 207} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0115571, size = 48, normalized size = 1.5 \[ -\frac{2 \sqrt{x} \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 37, normalized size = 1.2 \begin{align*} -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{\sqrt{x}\sqrt{cx+b}\sqrt{b}}{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \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{1}{\sqrt{c x^{2} + b x} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96808, size = 181, normalized size = 5.66 \begin{align*} \left [\frac{\log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right )}{\sqrt{b}}, \frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{b}\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{1}{\sqrt{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 = 1.24326, size = 53, normalized size = 1.66 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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