Optimal. Leaf size=75 \[ \frac{b (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{\sqrt{b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \]
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Rubi [A] time = 0.03355, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {779, 620, 206} \[ \frac{b (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{\sqrt{b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 779
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{\sqrt{b x+c x^2}} \, dx &=-\frac{(3 b B-4 A c-2 B c x) \sqrt{b x+c x^2}}{4 c^2}+\frac{(b (3 b B-4 A c)) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{8 c^2}\\ &=-\frac{(3 b B-4 A c-2 B c x) \sqrt{b x+c x^2}}{4 c^2}+\frac{(b (3 b B-4 A c)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{4 c^2}\\ &=-\frac{(3 b B-4 A c-2 B c x) \sqrt{b x+c x^2}}{4 c^2}+\frac{b (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0756727, size = 96, normalized size = 1.28 \[ \frac{b^{3/2} \sqrt{x} \sqrt{\frac{c x}{b}+1} (3 b B-4 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )+\sqrt{c} x (b+c x) (4 A c-3 b B+2 B c x)}{4 c^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 118, normalized size = 1.6 \begin{align*}{\frac{Bx}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,bB}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}B}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A}{c}\sqrt{c{x}^{2}+bx}}-{\frac{Ab}{2}\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 = 2.07077, size = 374, normalized size = 4.99 \begin{align*} \left [-\frac{{\left (3 \, B b^{2} - 4 \, A b c\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (2 \, B c^{2} x - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x}}{8 \, c^{3}}, -\frac{{\left (3 \, B b^{2} - 4 \, A b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (2 \, B c^{2} x - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x}}{4 \, c^{3}}\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{x \left (A + B x\right )}{\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 = 1.20914, size = 112, normalized size = 1.49 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, B x}{c} - \frac{3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac{{\left (3 \, B b^{2} - 4 \, A b c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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