Integrand size = 27, antiderivative size = 100 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=\frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 100, 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 = {828, 857, 635, 212, 738} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=-\frac {311 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{24} \sqrt {3 x^2+5 x+2} (73-6 x) \]
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{48} \int \frac {543+622 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311}{48} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {65}{8} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311}{24} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {65}{4} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = \frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.86 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=\frac {1}{72} \left (-3 (-73+6 x) \sqrt {2+5 x+3 x^2}+234 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-311 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
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Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (-73+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{24}-\frac {311 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{144}-\frac {13 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{8}\) | \(80\) |
trager | \(\left (\frac {73}{24}-\frac {x}{4}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{8}-\frac {311 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{144}\) | \(110\) |
default | \(-\frac {\left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{24}+\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{144}+\frac {13 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{8}-\frac {13 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{6}-\frac {13 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{8}\) | \(127\) |
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Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=-\frac {1}{24} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 73\right )} + \frac {311}{288} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac {13}{16} \, \sqrt {5} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=-\frac {1}{4} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {311}{144} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {13}{8} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {73}{24} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.26 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=-\frac {1}{24} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 73\right )} + \frac {13}{8} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {311}{144} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
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Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{2\,x+3} \,d x \]
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