Integrand size = 27, antiderivative size = 144 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40000 \sqrt {5}} \]
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Time = 0.05 (sec) , antiderivative size = 144, 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 = {848, 820, 734, 738, 212} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=-\frac {3159 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40000 \sqrt {5}}-\frac {87 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^3}-\frac {339 \left (3 x^2+5 x+2\right )^{3/2}}{500 (2 x+3)^4}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}+\frac {3159 (8 x+7) \sqrt {3 x^2+5 x+2}}{20000 (2 x+3)^2} \]
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Rule 212
Rule 734
Rule 738
Rule 820
Rule 848
Rubi steps \begin{align*} \text {integral}& = -\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {1}{25} \int \frac {\left (-\frac {105}{2}+78 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^5} \, dx \\ & = -\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}+\frac {1}{500} \int \frac {\left (\frac {2169}{2}-1017 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^4} \, dx \\ & = -\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac {3159 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1000} \\ & = \frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40000} \\ & = \frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac {3159 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20000} \\ & = \frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40000 \sqrt {5}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (244331+606326 x+549516 x^2+225816 x^3+35136 x^4\right )}{(3+2 x)^5}-3159 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{100000} \]
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Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\frac {105408 x^{6}+853128 x^{5}+2847900 x^{4}+5018190 x^{3}+4863655 x^{2}+2434307 x +488662}{20000 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+5 x +2}}+\frac {3159 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{200000}\) | \(83\) |
trager | \(\frac {\left (35136 x^{4}+225816 x^{3}+549516 x^{2}+606326 x +244331\right ) \sqrt {3 x^{2}+5 x +2}}{20000 \left (3+2 x \right )^{5}}-\frac {3159 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{200000}\) | \(92\) |
default | \(-\frac {339 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {87 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {3159 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {3159 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12500 \left (x +\frac {3}{2}\right )}-\frac {3159 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{200000}+\frac {3159 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{200000}+\frac {3159 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{25000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}\) | \(174\) |
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Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (35136 \, x^{4} + 225816 \, x^{3} + 549516 \, x^{2} + 606326 \, x + 244331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{400000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.47 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159}{200000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {9477}{20000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {339 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {87 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {3159 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3159 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{5000 \, {\left (2 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (118) = 236\).
Time = 0.32 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.52 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=-\frac {3159}{200000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {\sqrt {3} {\left (50544 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 2047032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11747352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 121295556 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 269183136 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1164571962 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1077361162 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 1845838971 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 592102521 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 244862928\right )}}{60000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
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Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^6} \,d x \]
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