Integrand size = 27, antiderivative size = 107 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=\frac {(121+124 x) \sqrt {2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac {1}{8} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {27 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{80 \sqrt {5}} \]
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Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {824, 857, 635, 212, 738} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=-\frac {1}{8} \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {27 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{80 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} (124 x+121)}{40 (2 x+3)^2} \]
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Rule 212
Rule 635
Rule 738
Rule 824
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {(121+124 x) \sqrt {2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac {1}{80} \int \frac {63+60 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {(121+124 x) \sqrt {2+5 x+3 x^2}}{40 (3+2 x)^2}+\frac {27}{80} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx-\frac {3}{8} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {(121+124 x) \sqrt {2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac {27}{40} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = \frac {(121+124 x) \sqrt {2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac {1}{8} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {27 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{80 \sqrt {5}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=\frac {1}{200} \left (\frac {5 (121+124 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+27 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-50 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
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Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {372 x^{3}+983 x^{2}+853 x +242}{40 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{8}-\frac {27 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{400}\) | \(97\) |
trager | \(\frac {\left (121+124 x \right ) \sqrt {3 x^{2}+5 x +2}}{40 \left (3+2 x \right )^{2}}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{400}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{8}\) | \(118\) |
default | \(-\frac {21 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{50 \left (x +\frac {3}{2}\right )}+\frac {27 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{400}-\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{8}-\frac {27 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{400}+\frac {21 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{100}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}\) | \(142\) |
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Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.34 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=\frac {50 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 27 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (124 \, x + 121\right )}}{800 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.22 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=-\frac {1}{8} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {27}{400} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {39}{40} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {21 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{20 \, {\left (2 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (83) = 166\).
Time = 0.32 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.24 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=\frac {27}{400} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {1}{8} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {886 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 2897 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 9039 \, \sqrt {3} x + 3037 \, \sqrt {3} - 9039 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{40 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]
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Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^3} \,d x \]
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