Optimal. Leaf size=110 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac{2267 (6 x+5) \sqrt{3 x^2+5 x+2}}{2592}-\frac{2267 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5184 \sqrt{3}} \]
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Rubi [A] time = 0.0488296, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac{2267 (6 x+5) \sqrt{3 x^2+5 x+2}}{2592}-\frac{2267 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5184 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^2 \sqrt{2+5 x+3 x^2} \, dx &=-\frac{1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{1}{15} \int (3+2 x) \left (\frac{511}{2}+167 x\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=-\frac{1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}+\frac{2267}{216} \int \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{2267 (5+6 x) \sqrt{2+5 x+3 x^2}}{2592}-\frac{1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac{2267 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{5184}\\ &=\frac{2267 (5+6 x) \sqrt{2+5 x+3 x^2}}{2592}-\frac{1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac{2267 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{2592}\\ &=\frac{2267 (5+6 x) \sqrt{2+5 x+3 x^2}}{2592}-\frac{1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac{2267 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{5184 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.037197, size = 72, normalized size = 0.65 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (10368 x^4-23760 x^3-229416 x^2-375250 x-168627\right )-11335 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{77760} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 96, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{19\,x}{18} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{6997}{1620} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{11335+13602\,x}{2592}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{2267\,\sqrt{3}}{15552}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49892, size = 140, normalized size = 1.27 \begin{align*} -\frac{4}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{19}{18} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{6997}{1620} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{2267}{432} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{2267}{15552} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{11335}{2592} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27092, size = 243, normalized size = 2.21 \begin{align*} -\frac{1}{12960} \,{\left (10368 \, x^{4} - 23760 \, x^{3} - 229416 \, x^{2} - 375250 \, x - 168627\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{2267}{31104} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\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 - 51 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 8 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 4 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 45 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19768, size = 93, normalized size = 0.85 \begin{align*} -\frac{1}{12960} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (24 \, x - 55\right )} x - 9559\right )} x - 187625\right )} x - 168627\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{2267}{15552} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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