Optimal. Leaf size=134 \[ -\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.0551097, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {670, 640, 612, 620, 206} \[ -\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}-\frac{(7 b) \int x \left (b x+c x^2\right )^{3/2} \, dx}{12 c}\\ &=-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (7 b^2\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c^2}\\ &=\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (7 b^4\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (7 b^6\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (7 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{7 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.167331, size = 120, normalized size = 0.9 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-56 b^3 c^2 x^2+48 b^2 c^3 x^3+70 b^4 c x-105 b^5+1664 b c^4 x^4+1280 c^5 x^5\right )+\frac{105 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{7680 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 146, normalized size = 1.1 \begin{align*}{\frac{x}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,b}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{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 = 1.94666, size = 514, normalized size = 3.84 \begin{align*} \left [\frac{105 \, b^{6} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} + 48 \, b^{2} c^{4} x^{3} - 56 \, b^{3} c^{3} x^{2} + 70 \, b^{4} c^{2} x - 105 \, b^{5} c\right )} \sqrt{c x^{2} + b x}}{15360 \, c^{5}}, -\frac{105 \, b^{6} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} + 48 \, b^{2} c^{4} x^{3} - 56 \, b^{3} c^{3} x^{2} + 70 \, b^{4} c^{2} x - 105 \, b^{5} c\right )} \sqrt{c x^{2} + b x}}{7680 \, 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} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34954, size = 146, normalized size = 1.09 \begin{align*} -\frac{7 \, b^{6} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} + \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x + 13 \, b\right )} x + \frac{3 \, b^{2}}{c}\right )} x - \frac{7 \, b^{3}}{c^{2}}\right )} x + \frac{35 \, b^{4}}{c^{3}}\right )} x - \frac{105 \, b^{5}}{c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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