Integrand size = 27, antiderivative size = 197 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {603}{512} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {934161 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{204800 \sqrt {5}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {824, 826, 857, 635, 212, 738} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {603}{512} \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {934161 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{204800 \sqrt {5}}+\frac {(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac {3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac {(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac {3 (61278 x+131465) \sqrt {3 x^2+5 x+2}}{102400 (2 x+3)} \]
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 824
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac {1}{240} \int \frac {(513+522 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx \\ & = \frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {\int \frac {(-78210-91260 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{38400} \\ & = \frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac {\int \frac {(9426420+11030040 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{3072000} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {\int \frac {148396680+173664000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{24576000} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {1809}{512} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {934161 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{204800} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {1809}{256} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {934161 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{102400} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {603}{512} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {934161 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{204800 \sqrt {5}} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (1810375853+7753535702 x+13975079520 x^2+13619671040 x^3+7622049520 x^4+2361590432 x^5+338443008 x^6+9676800 x^7\right )}{(3+2 x)^7}-6539127 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+8442000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{3584000} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {29030400 x^{9}+1063713024 x^{8}+8796339936 x^{7}+35350986736 x^{6}+83692441584 x^{5}+125267692800 x^{4}+120375346786 x^{3}+72148965109 x^{2}+24558950669 x +3620751706}{716800 \left (3+2 x \right )^{7} \sqrt {3 x^{2}+5 x +2}}+\frac {603 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{512}+\frac {934161 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024000}\) | \(127\) |
trager | \(-\frac {\left (9676800 x^{7}+338443008 x^{6}+2361590432 x^{5}+7622049520 x^{4}+13619671040 x^{3}+13975079520 x^{2}+7753535702 x +1810375853\right ) \sqrt {3 x^{2}+5 x +2}}{716800 \left (3+2 x \right )^{7}}+\frac {603 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{512}-\frac {934161 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1024000}\) | \(148\) |
default | \(-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{896 \left (x +\frac {3}{2}\right )^{6}}-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{500 \left (x +\frac {3}{2}\right )^{5}}-\frac {4719 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{560000 \left (x +\frac {3}{2}\right )^{4}}-\frac {5147 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{350000 \left (x +\frac {3}{2}\right )^{3}}-\frac {15267 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {78423 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1750000}-\frac {78423 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{875000 \left (x +\frac {3}{2}\right )}+\frac {47541 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000000}+\frac {14777 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{160000}+\frac {29661 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{128000}+\frac {603 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{512}+\frac {934161 \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 )}{1024000}-\frac {934161 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4000000}-\frac {311387 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{640000}-\frac {934161 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{7000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {934161 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024000}\) | \(358\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.26 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {8442000 \, \sqrt {3} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 6539127 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (9676800 \, x^{7} + 338443008 \, x^{6} + 2361590432 \, x^{5} + 7622049520 \, x^{4} + 13619671040 \, x^{3} + 13975079520 \, x^{2} + 7753535702 \, x + 1810375853\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{14336000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
[In]
[Out]
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (161) = 322\).
Time = 0.28 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.15 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {45801}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{14 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {24 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {4719 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{35000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {5147 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{43750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {15267 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {142623}{500000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {16659}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {78423 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{350000 \, {\left (2 \, x + 3\right )}} + \frac {44331}{80000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {15847}{640000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {88983}{64000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {603}{512} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {934161}{1024000} \, \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 {340941}{512000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (161) = 322\).
Time = 0.36 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.58 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\frac {934161}{1024000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {603}{512} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {27}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2310353472 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 39459777504 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 930047331808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 4439192854544 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 42996771835920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 98991221694624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 500967391220544 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 626374342937616 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 1740466332835804 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1179088946690970 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1703610278292706 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 552456024942507 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 324453464706399 \, \sqrt {3} x + 28970271150072 \, \sqrt {3} - 324453464706399 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{716800 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]
[In]
[Out]
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^8} \,d x \]
[In]
[Out]