Integrand size = 27, antiderivative size = 124 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25 \sqrt {5}} \]
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Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {836, 820, 738, 212} \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {408 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25 \sqrt {5}}-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4416 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)}+\frac {4 (462 x+401)}{5 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
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Rule 212
Rule 738
Rule 820
Rule 836
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {1029+846 x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {12510+13860 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408}{25} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}-\frac {816}{25} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = -\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25 \sqrt {5}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.73 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2}{125} \left (\frac {5 \sqrt {2+5 x+3 x^2} \left (16667+73215 x+116826 x^2+80100 x^3+19872 x^4\right )}{(1+x)^2 (3+2 x) (2+3 x)^2}+408 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right ) \]
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Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {\frac {39744}{25} x^{4}+6408 x^{3}+\frac {233652}{25} x^{2}+\frac {29286}{5} x +\frac {33334}{25}}{\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {408 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{125}\) | \(73\) |
trager | \(\frac {\frac {39744}{25} x^{4}+6408 x^{3}+\frac {233652}{25} x^{2}+\frac {29286}{5} x +\frac {33334}{25}}{\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {408 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{125}\) | \(92\) |
default | \(\frac {17}{5 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {16 \left (5+6 x \right )}{5 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {1104}{5}+\frac {6624 x}{25}}{\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {204}{25 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {408 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{125}-\frac {13}{10 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(127\) |
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Time = 0.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.13 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (102 \, \sqrt {5} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (19872 \, x^{4} + 80100 \, x^{3} + 116826 \, x^{2} + 73215 \, x + 16667\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{125 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \]
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\[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.09 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {408}{125} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {6624 \, x}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {5724}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {96 \, x}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{5 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {63}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (102) = 204\).
Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.90 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {24}{125} \, \sqrt {5} {\left (92 \, \sqrt {5} \sqrt {3} - 17 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {408 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{125 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {8 \, {\left (\frac {\frac {\frac {5 \, {\left (\frac {972}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {13}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {12324}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {9783}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2484}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{25 \, {\left (\frac {8}{2 \, x + 3} - \frac {5}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} \]
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Timed out. \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^2\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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