Integrand size = 27, antiderivative size = 169 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {26453 (7+8 x) \sqrt {2+5 x+3 x^2}}{200000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}-\frac {26453 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{400000 \sqrt {5}} \]
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Time = 0.07 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {848, 820, 734, 738, 212} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=-\frac {26453 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{400000 \sqrt {5}}-\frac {2237 \left (3 x^2+5 x+2\right )^{3/2}}{3750 (2 x+3)^3}-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{5000 (2 x+3)^4}-\frac {73 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^5}-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}+\frac {26453 (8 x+7) \sqrt {3 x^2+5 x+2}}{200000 (2 x+3)^2} \]
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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}}{30 (3+2 x)^6}-\frac {1}{30} \int \frac {\left (-\frac {87}{2}+117 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx \\ & = -\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}+\frac {1}{750} \int \frac {\left (\frac {1455}{2}-2628 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}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {\int \frac {\left (-\frac {50169}{2}+28017 x\right ) \sqrt {2+5 x+3 x^2}}{(3+2 x)^4} \, dx}{15000} \\ & = -\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}+\frac {26453 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{10000} \\ & = \frac {26453 (7+8 x) \sqrt {2+5 x+3 x^2}}{200000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}-\frac {26453 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{400000} \\ & = \frac {26453 (7+8 x) \sqrt {2+5 x+3 x^2}}{200000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}+\frac {26453 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{200000} \\ & = \frac {26453 (7+8 x) \sqrt {2+5 x+3 x^2}}{200000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}-\frac {26453 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{400000 \sqrt {5}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (16322393+50707640 x+62797200 x^2+39304480 x^3+12381040 x^4+1567872 x^5\right )}{(3+2 x)^6}-79359 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{3000000} \]
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Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {4703616 x^{7}+44982480 x^{6}+182954384 x^{5}+409676080 x^{4}+544717880 x^{3}+428099779 x^{2}+183027245 x +32644786}{600000 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+5 x +2}}+\frac {26453 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{2000000}\) | \(88\) |
trager | \(\frac {\left (1567872 x^{5}+12381040 x^{4}+39304480 x^{3}+62797200 x^{2}+50707640 x +16322393\right ) \sqrt {3 x^{2}+5 x +2}}{600000 \left (3+2 x \right )^{6}}-\frac {26453 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{2000000}\) | \(97\) |
default | \(-\frac {73 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4000 \left (x +\frac {3}{2}\right )^{5}}-\frac {3113 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2237 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {26453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{200000 \left (x +\frac {3}{2}\right )^{2}}-\frac {26453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {26453 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{2000000}+\frac {26453 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{2000000}+\frac {26453 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{250000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}\) | \(195\) |
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Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {79359 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1567872 \, x^{5} + 12381040 \, x^{4} + 39304480 \, x^{3} + 62797200 \, x^{2} + 50707640 \, x + 16322393\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{12000000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.53 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {26453}{2000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {79359}{200000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {73 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {3113 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {2237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {26453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{50000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {26453 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{50000 \, {\left (2 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (139) = 278\).
Time = 0.33 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.43 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=-\frac {26453}{2000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2539488 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 41901552 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 924796880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 3988893600 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 33933192480 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 66530947296 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 275158218192 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 265623867480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 526452161650 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 226453420305 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 171288605499 \, \sqrt {3} x + 19197814536 \, \sqrt {3} - 171288605499 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{600000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]
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Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^7} \,d x \]
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