Optimal. Leaf size=60 \[ \frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{3/2}} \]
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Rubi [A] time = 0.0136197, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 620, 206} \[ \frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \sqrt{b x+c x^2} \, dx &=\frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{8 c}\\ &=\frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{4 c}\\ &=\frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0999567, size = 74, normalized size = 1.23 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} (b+2 c x)-\frac{b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 56, normalized size = 0.9 \begin{align*}{\frac{2\,cx+b}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \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 = 2.048, size = 285, normalized size = 4.75 \begin{align*} \left [\frac{b^{2} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x}}{8 \, c^{2}}, \frac{b^{2} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x}}{4 \, c^{2}}\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 \sqrt{b x + c x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40028, size = 82, normalized size = 1.37 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (2 \, x + \frac{b}{c}\right )} + \frac{b^{2} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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