Optimal. Leaf size=97 \[ \frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}-\frac{(b+2 c x) \sqrt{b x+c x^2} (b B-2 A c)}{8 c^2}+\frac{B \left (b x+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.0392168, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {640, 612, 620, 206} \[ \frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}-\frac{(b+2 c x) \sqrt{b x+c x^2} (b B-2 A c)}{8 c^2}+\frac{B \left (b x+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac{(-b B+2 A c) \int \sqrt{b x+c x^2} \, dx}{2 c}\\ &=-\frac{(b B-2 A c) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac{\left (b^2 (b B-2 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c^2}\\ &=-\frac{(b B-2 A c) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac{\left (b^2 (b B-2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c^2}\\ &=-\frac{(b B-2 A c) (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.202767, size = 108, normalized size = 1.11 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (2 b c (3 A+B x)+4 c^2 x (3 A+2 B x)-3 b^2 B\right )+\frac{3 b^{3/2} (b B-2 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{24 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 157, normalized size = 1.6 \begin{align*}{\frac{B}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bBx}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}B}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{3}B}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Ax}{2}\sqrt{c{x}^{2}+bx}}+{\frac{Ab}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{A{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 = 1.35495, size = 468, normalized size = 4.82 \begin{align*} \left [-\frac{3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (8 \, B c^{3} x^{2} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{48 \, c^{3}}, -\frac{3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, B c^{3} x^{2} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{24 \, 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 \sqrt{x \left (b + c x\right )} \left (A + B x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24815, size = 138, normalized size = 1.42 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, B x + \frac{B b c + 6 \, A c^{2}}{c^{2}}\right )} x - \frac{3 \,{\left (B b^{2} - 2 \, A b c\right )}}{c^{2}}\right )} - \frac{{\left (B b^{3} - 2 \, A b^{2} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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