Optimal. Leaf size=123 \[ \frac{1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{128} (175-414 x) \sqrt{3 x^2+5 x+2}-\frac{2011 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}+\frac{65}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.0789957, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {814, 843, 621, 206, 724} \[ \frac{1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{128} (175-414 x) \sqrt{3 x^2+5 x+2}-\frac{2011 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}+\frac{65}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
Antiderivative was successfully verified.
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Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx &=\frac{1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{96} \int \frac{(1083+1242 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\\ &=\frac{1}{128} (175-414 x) \sqrt{2+5 x+3 x^2}+\frac{1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac{\int \frac{-61794-72396 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{4608}\\ &=\frac{1}{128} (175-414 x) \sqrt{2+5 x+3 x^2}+\frac{1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2011}{256} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{325}{32} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{128} (175-414 x) \sqrt{2+5 x+3 x^2}+\frac{1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2011}{128} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{325}{16} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{1}{128} (175-414 x) \sqrt{2+5 x+3 x^2}+\frac{1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2011 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{256 \sqrt{3}}+\frac{65}{32} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0477458, size = 103, normalized size = 0.84 \[ \frac{1}{768} \left (-2 \sqrt{3 x^2+5 x+2} \left (144 x^3-888 x^2-542 x-1277\right )-1560 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-2011 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 183, normalized size = 1.5 \begin{align*} -{\frac{5+6\,x}{48} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5+6\,x}{384}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{\sqrt{3}}{2304}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{65+78\,x}{24}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{377\,\sqrt{3}}{144}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{65}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{65\,\sqrt{5}}{32}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50026, size = 173, normalized size = 1.41 \begin{align*} -\frac{1}{8} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{47}{48} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{207}{64} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{2011}{768} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{65}{32} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{175}{128} \, \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 = 1.5335, size = 354, normalized size = 2.88 \begin{align*} -\frac{1}{384} \,{\left (144 \, x^{3} - 888 \, x^{2} - 542 \, x - 1277\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{2011}{1536} \, \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 ) + \frac{65}{64} \, \sqrt{5} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\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 - \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23649, size = 184, normalized size = 1.5 \begin{align*} -\frac{1}{384} \,{\left (2 \,{\left (12 \,{\left (6 \, x - 37\right )} x - 271\right )} x - 1277\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{65}{32} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2011}{768} \, \sqrt{3} \log \left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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