Integrand size = 24, antiderivative size = 180 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=-\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{735306250 \sqrt {35}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {849, 821, 735, 739, 212} \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=-\frac {76869 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{735306250 \sqrt {35}}-\frac {4741 \left (3 x^2+2\right )^{7/2}}{1800750 (2 x+3)^7}-\frac {27 \left (3 x^2+2\right )^{7/2}}{2450 (2 x+3)^8}-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac {949 (4-9 x) \left (3 x^2+2\right )^{5/2}}{3001250 (2 x+3)^6}-\frac {2847 (4-9 x) \left (3 x^2+2\right )^{3/2}}{42017500 (2 x+3)^4}-\frac {25623 (4-9 x) \sqrt {3 x^2+2}}{1470612500 (2 x+3)^2} \]
[In]
[Out]
Rule 212
Rule 735
Rule 739
Rule 821
Rule 849
Rubi steps \begin{align*} \text {integral}& = -\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {1}{315} \int \frac {(-369+78 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx \\ & = -\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}+\frac {\int \frac {(24072-2916 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{88200} \\ & = -\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {2847 \int \frac {\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{42875} \\ & = -\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {2847 \int \frac {\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{300125} \\ & = -\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {25623 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{21008750} \\ & = -\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}+\frac {76869 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{735306250} \\ & = -\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{735306250} \\ & = -\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{735306250 \sqrt {35}} \\ \end{align*}
Time = 2.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.60 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (15948113036+11990965797 x+42455611758 x^2-11567526201 x^3+9750269970 x^4-25197346566 x^5-620594352 x^6+30006612 x^7+10968696 x^8\right )}{(3+2 x)^9}+2767284 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{463242937500} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56
method | result | size |
risch | \(-\frac {32906088 x^{10}+90019836 x^{9}-1839845664 x^{8}-75532026474 x^{7}+28009621206 x^{6}-85097271735 x^{5}+146867375214 x^{4}+12837844989 x^{3}+132755562624 x^{2}+23981931594 x +31896226072}{13235512500 \left (3+2 x \right )^{9} \sqrt {3 x^{2}+2}}-\frac {76869 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{25735718750}\) | \(100\) |
trager | \(-\frac {\left (10968696 x^{8}+30006612 x^{7}-620594352 x^{6}-25197346566 x^{5}+9750269970 x^{4}-11567526201 x^{3}+42455611758 x^{2}+11990965797 x +15948113036\right ) \sqrt {3 x^{2}+2}}{13235512500 \left (3+2 x \right )^{9}}-\frac {76869 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{25735718750}\) | \(106\) |
default | \(\frac {102492 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{450375078125}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{161280 \left (x +\frac {3}{2}\right )^{9}}-\frac {27 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{627200 \left (x +\frac {3}{2}\right )^{8}}-\frac {4741 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{230496000 \left (x +\frac {3}{2}\right )^{7}}-\frac {949 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{96040000 \left (x +\frac {3}{2}\right )^{6}}-\frac {8541 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1680700000 \left (x +\frac {3}{2}\right )^{5}}-\frac {82563 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{29412250000 \left (x +\frac {3}{2}\right )^{4}}-\frac {845559 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{514714375000 \left (x +\frac {3}{2}\right )^{3}}-\frac {9198657 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9007501562500 \left (x +\frac {3}{2}\right )^{2}}+\frac {320313123 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{157631277343750}-\frac {106771041 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{157631277343750 \left (x +\frac {3}{2}\right )}+\frac {8993673 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1801500312500}+\frac {691821 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{51471437500}-\frac {76869 \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 )}{25735718750}+\frac {76869 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{25735718750}+\frac {1229904 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{78815638671875}\) | \(320\) |
[In]
[Out]
none
Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.08 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {691821 \, \sqrt {35} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (10968696 \, x^{8} + 30006612 \, x^{7} - 620594352 \, x^{6} - 25197346566 \, x^{5} + 9750269970 \, x^{4} - 11567526201 \, x^{3} + 42455611758 \, x^{2} + 11990965797 \, x + 15948113036\right )} \sqrt {3 \, x^{2} + 2}}{463242937500 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (149) = 298\).
Time = 0.30 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.41 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {27595971}{9007501562500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{315 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2450 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {4741 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1800750 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {949 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1500625 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {8541 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{52521875 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {82563 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1838265625 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {845559 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{64339296875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9198657 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2251875390625 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {8993673}{1801500312500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {102492}{450375078125} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {106771041 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{9007501562500 \, {\left (2 \, x + 3\right )}} + \frac {691821}{51471437500} \, \sqrt {3 \, x^{2} + 2} x + \frac {76869}{25735718750} \, \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 {76869}{12867859375} \, \sqrt {3 \, x^{2} + 2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (149) = 298\).
Time = 0.33 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.79 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {76869}{25735718750} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, \sqrt {3} {\left (364416 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{17} + 27877824 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{16} + 1042205258 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} - 956098170 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} + 1003625490 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} - 85987901496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 60468401868 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 331045664193 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 22913148915 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 544736640510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 284856270864 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 908850124224 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 90616216992 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 115517223360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 52895204480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 565618176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 140708352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 17333248\right )}}{94119200000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{9}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.14 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {76869\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{25735718750}-\frac {76869\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{25735718750}+\frac {4515\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32768\,\left (x^8+12\,x^7+63\,x^6+189\,x^5+\frac {2835\,x^4}{8}+\frac {1701\,x^3}{4}+\frac {5103\,x^2}{16}+\frac {2187\,x}{16}+\frac {6561}{256}\right )}+\frac {1838301\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{614656000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{294912\,\left (x^9+\frac {27\,x^8}{2}+81\,x^7+\frac {567\,x^6}{2}+\frac {5103\,x^5}{8}+\frac {15309\,x^4}{16}+\frac {15309\,x^3}{16}+\frac {19683\,x^2}{32}+\frac {59049\,x}{256}+\frac {19683}{512}\right )}-\frac {923241\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{35123200\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {152343\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{94119200000\,\left (x+\frac {3}{2}\right )}+\frac {35213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{401408\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {80649\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{5378240000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {52201\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{344064\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {55473\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536640000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
[In]
[Out]