Integrand size = 27, antiderivative size = 124 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{32000 \sqrt {5}} \]
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Time = 0.04 (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 = {820, 734, 738, 212} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {141 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{32000 \sqrt {5}}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {141 (8 x+7) \sqrt {3 x^2+5 x+2}}{16000 (2 x+3)^2} \]
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Rule 212
Rule 734
Rule 738
Rule 820
Rubi steps \begin{align*} \text {integral}& = -\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {47}{10} \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx \\ & = \frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141}{800} \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx \\ & = -\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{32000} \\ & = -\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{16000} \\ & = -\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{32000 \sqrt {5}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.63 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (19031+90126 x+131516 x^2+66616 x^3+6336 x^4\right )}{(3+2 x)^5}+141 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{80000} \]
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Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {19008 x^{6}+231528 x^{5}+740300 x^{4}+1061190 x^{3}+770755 x^{2}+275407 x +38062}{16000 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+5 x +2}}-\frac {141 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{160000}\) | \(83\) |
trager | \(\frac {\left (6336 x^{4}+66616 x^{3}+131516 x^{2}+90126 x +19031\right ) \sqrt {3 x^{2}+5 x +2}}{16000 \left (3+2 x \right )^{5}}+\frac {141 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{160000}\) | \(92\) |
default | \(-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1600 \left (x +\frac {3}{2}\right )^{4}}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1457 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {1363 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{12500 \left (x +\frac {3}{2}\right )}+\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{100000}-\frac {141 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{20000}+\frac {141 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{160000}-\frac {141 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{160000}+\frac {1363 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{25000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}\) | \(211\) |
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Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {141 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (6336 \, x^{4} + 66616 \, x^{3} + 131516 \, x^{2} + 90126 \, x + 19031\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{320000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (102) = 204\).
Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.94 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {4371}{20000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{100 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {423}{10000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {141}{160000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {2679}{80000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (2 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (102) = 204\).
Time = 0.32 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.90 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {141}{160000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {146256 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 654456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 415048 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 15455452 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 140042336 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 207568854 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 544555762 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 286352757 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 252454821 \, \sqrt {3} x - 31985676 \, \sqrt {3} + 252454821 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^6} \,d x \]
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