Integrand size = 24, antiderivative size = 133 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {159759 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{219520 \sqrt {35}} \]
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Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {825, 858, 221, 739, 212} \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=-\frac {9}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {159759 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{219520 \sqrt {35}}+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{1568 (2 x+3)^4}+\frac {9 (5167 x+4373) \sqrt {3 x^2+2}}{109760 (2 x+3)^2} \]
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Rule 212
Rule 221
Rule 739
Rule 825
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {\int \frac {(-1560+1260 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{1680} \\ & = \frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}+\frac {\int \frac {(496800-1058400 x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{1881600} \\ & = \frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {\int \frac {-100051200+444528000 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1053696000} \\ & = \frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {27}{128} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {159759 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{219520} \\ & = \frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {159759 \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{219520} \\ & = \frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {159759 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{219520 \sqrt {35}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {\frac {70 \sqrt {2+3 x^2} \left (10361807+39843609 x+59256588 x^2+47453802 x^3+18915336 x^4+4369608 x^5\right )}{(3+2 x)^6}+958554 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+1620675 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{23049600} \]
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Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {13108824 x^{7}+56746008 x^{6}+151100622 x^{5}+215600436 x^{4}+214438431 x^{3}+149598597 x^{2}+79687218 x +20723614}{329280 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+2}}-\frac {9 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}-\frac {159759 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7683200}\) | \(97\) |
trager | \(\frac {\left (4369608 x^{5}+18915336 x^{4}+47453802 x^{3}+59256588 x^{2}+39843609 x +10361807\right ) \sqrt {3 x^{2}+2}}{329280 \left (3+2 x \right )^{6}}-\frac {159759 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{7683200}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}\) | \(121\) |
default | \(-\frac {9 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}+\frac {53253 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{33614000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {\left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{3136 \left (x +\frac {3}{2}\right )^{5}}-\frac {113 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{548800 \left (x +\frac {3}{2}\right )^{4}}-\frac {1039 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9604000 \left (x +\frac {3}{2}\right )^{3}}-\frac {6561 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{84035000 \left (x +\frac {3}{2}\right )^{2}}+\frac {123129 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1470612500}-\frac {41043 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1470612500 \left (x +\frac {3}{2}\right )}-\frac {27009 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{67228000}-\frac {45711 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{3841600}-\frac {159759 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7683200}+\frac {159759 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{7683200}+\frac {159759 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1470612500}\) | \(269\) |
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Time = 0.39 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {1620675 \, \sqrt {3} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 479277 \, \sqrt {35} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 140 \, {\left (4369608 \, x^{5} + 18915336 \, x^{4} + 47453802 \, x^{3} + 59256588 \, x^{2} + 39843609 \, x + 10361807\right )} \sqrt {3 \, x^{2} + 2}}{46099200 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (107) = 214\).
Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.16 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {19683}{84035000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {{\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{98 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {113 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{34300 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1039 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1200500 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {6561 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{21008750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {27009}{67228000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {53253}{33614000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {41043 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{84035000 \, {\left (2 \, x + 3\right )}} - \frac {45711}{3841600} \, \sqrt {3 \, x^{2} + 2} x - \frac {9}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {159759}{7683200} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {159759}{3841600} \, \sqrt {3 \, x^{2} + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (107) = 214\).
Time = 0.31 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {9}{128} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {159759}{7683200} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, \sqrt {3} {\left (566976 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 16427322 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 70792520 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 421378065 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 244013814 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 879808433 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 512612604 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 2079633300 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 831934400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 500387712 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 51770496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 7768192\right )}}{878080 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \]
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Time = 0.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.79 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {159759\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{7683200}-\frac {9\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {159759\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7683200}-\frac {9019\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {7315\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}+\frac {182067\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{878080\,\left (x+\frac {3}{2}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{24576\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}-\frac {164961\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{250880\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {109789\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{71680\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
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