Optimal. Leaf size=165 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} (7 b B-10 A c)}{128 c^4}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}+\frac{b \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{48 c^3}-\frac{x \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
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Rubi [A] time = 0.159874, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {794, 670, 640, 612, 620, 206} \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} (7 b B-10 A c)}{128 c^4}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}+\frac{b \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{48 c^3}-\frac{x \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 670
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x^2 (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (2 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \int x^2 \sqrt{b x+c x^2} \, dx}{5 c}\\ &=-\frac{(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{(b (7 b B-10 A c)) \int x \sqrt{b x+c x^2} \, dx}{16 c^2}\\ &=\frac{b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac{\left (b^2 (7 b B-10 A c)\right ) \int \sqrt{b x+c x^2} \, dx}{32 c^3}\\ &=-\frac{b^2 (7 b B-10 A c) (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (b^4 (7 b B-10 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^4}\\ &=-\frac{b^2 (7 b B-10 A c) (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (b^4 (7 b B-10 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{128 c^4}\\ &=-\frac{b^2 (7 b B-10 A c) (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.312663, size = 148, normalized size = 0.9 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-4 b^2 c^2 x (25 A+14 B x)+10 b^3 c (15 A+7 B x)+16 b c^3 x^2 (5 A+3 B x)+96 c^4 x^3 (5 A+4 B x)-105 b^4 B\right )+\frac{15 b^{7/2} (7 b B-10 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{1920 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 245, normalized size = 1.5 \begin{align*}{\frac{B{x}^{2}}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,bBx}{40\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}B}{48\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}Bx}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{b}^{4}B}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,B{b}^{5}}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{Ax}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ab}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,A{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{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 = 2.10016, size = 709, normalized size = 4.3 \begin{align*} \left [-\frac{15 \,{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \,{\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \,{\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \,{\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{3840 \, c^{5}}, -\frac{15 \,{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \,{\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \,{\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \,{\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{1920 \, c^{5}}\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 x^{2} \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 = 1.16774, size = 216, normalized size = 1.31 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B x + \frac{B b c^{3} + 10 \, A c^{4}}{c^{4}}\right )} x - \frac{7 \, B b^{2} c^{2} - 10 \, A b c^{3}}{c^{4}}\right )} x + \frac{5 \,{\left (7 \, B b^{3} c - 10 \, A b^{2} c^{2}\right )}}{c^{4}}\right )} x - \frac{15 \,{\left (7 \, B b^{4} - 10 \, A b^{3} c\right )}}{c^{4}}\right )} - \frac{{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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