Optimal. Leaf size=105 \[ \frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.0374605, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]
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 \sqrt{b x+c x^2} \, dx &=\frac{x \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{(5 b) \int x \sqrt{b x+c x^2} \, dx}{8 c}\\ &=-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c}+\frac{\left (5 b^2\right ) \int \sqrt{b x+c x^2} \, dx}{16 c^2}\\ &=\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (5 b^4\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^3}\\ &=\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^3}\\ &=\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.13163, size = 98, normalized size = 0.93 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-10 b^2 c x+15 b^3+8 b c^2 x^2+48 c^3 x^3\right )-\frac{15 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 107, normalized size = 1. \begin{align*}{\frac{x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{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 = 2.0657, size = 398, normalized size = 3.79 \begin{align*} \left [\frac{15 \, b^{4} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} - 10 \, b^{2} c^{2} x + 15 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{384 \, c^{4}}, \frac{15 \, b^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} - 10 \, b^{2} c^{2} x + 15 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{192 \, c^{4}}\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 )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35702, size = 115, normalized size = 1.1 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x + \frac{b}{c}\right )} x - \frac{5 \, b^{2}}{c^{2}}\right )} x + \frac{15 \, b^{3}}{c^{3}}\right )} + \frac{5 \, b^{4} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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