Optimal. Leaf size=47 \[ 2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \sqrt{b x+c x^2}}{x} \]
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Rubi [A] time = 0.0177092, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {662, 620, 206} \[ 2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \sqrt{b x+c x^2}}{x} \]
Antiderivative was successfully verified.
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Rule 662
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{x^2} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{x}+c \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 \sqrt{b x+c x^2}}{x}+(2 c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=-\frac{2 \sqrt{b x+c x^2}}{x}+2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0839122, size = 63, normalized size = 1.34 \[ \frac{2 \sqrt{x (b+c x)} \left (\frac{\sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}-1\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 66, normalized size = 1.4 \begin{align*} -2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{3/2}}{b{x}^{2}}}+2\,{\frac{c\sqrt{c{x}^{2}+bx}}{b}}+\sqrt{c}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \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 = 2.02655, size = 224, normalized size = 4.77 \begin{align*} \left [\frac{\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}}{x}, -\frac{2 \,{\left (\sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}\right )}}{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{\sqrt{x \left (b + c x\right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37555, size = 81, normalized size = 1.72 \begin{align*} -\sqrt{c} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \, b}{\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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