Optimal. Leaf size=133 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac{(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac{1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{1129 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{1129 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]
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Rubi [A] time = 0.0612113, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac{(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac{1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{1129 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{1129 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]
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 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{1}{21} \int (3+2 x) \left (\frac{721}{2}+237 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac{1129}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}-\frac{1129 \int \sqrt{2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac{1129 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac{1129 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac{1129 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac{1129 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac{1129 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac{1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac{1129 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{41472 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0596527, size = 82, normalized size = 0.62 \[ \frac{39515 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (1244160 x^6-311040 x^5-27084672 x^4-79049520 x^3-94861176 x^2-51971350 x-10669737\right )}{4354560} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 115, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{21} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{43\,x}{63} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5017}{1890} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5645+6774\,x}{2592} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5645+6774\,x}{20736}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1129\,\sqrt{3}}{124416}\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 = 1.53682, size = 180, normalized size = 1.35 \begin{align*} -\frac{4}{21} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{43}{63} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{5017}{1890} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1129}{432} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{5645}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{1129}{3456} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1129}{124416} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{5645}{20736} \, \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 = 1.35341, size = 297, normalized size = 2.23 \begin{align*} -\frac{1}{725760} \,{\left (1244160 \, x^{6} - 311040 \, x^{5} - 27084672 \, x^{4} - 79049520 \, x^{3} - 94861176 \, x^{2} - 51971350 \, x - 10669737\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1129}{248832} \, \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 - 327 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 406 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 185 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 4 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 12 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 90 \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 = 1.17889, size = 107, normalized size = 0.8 \begin{align*} -\frac{1}{725760} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (90 \,{\left (4 \, x - 1\right )} x - 7837\right )} x - 182985\right )} x - 3952549\right )} x - 25985675\right )} x - 10669737\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1129}{124416} \, \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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