Optimal. Leaf size=209 \[ -\frac {29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac {13251 (8 x+7) \sqrt {3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac {13251 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2048000000 \sqrt {5}} \]
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Rubi [A] time = 0.12, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \begin {gather*} -\frac {29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac {13251 (8 x+7) \sqrt {3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac {13251 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2048000000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {1}{50} \int \frac {\left (-\frac {405}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {1893}{500} \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{80000}\\ &=-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {4417 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{640000}\\ &=\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{51200000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {13251 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2048000000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1024000000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {13251 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2048000000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 212, normalized size = 1.01 \begin {gather*} \frac {1}{50} \left (-\frac {58 \left (3 x^2+5 x+2\right )^{9/2}}{5 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {4417 \left (\frac {32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac {2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac {3 (8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac {3 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )}{1024000}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.94, size = 106, normalized size = 0.51 \begin {gather*} \frac {13251 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{1024000000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (371791872 x^9+5268182272 x^8+40186580992 x^7+148740043392 x^6+304078211712 x^5+372602220928 x^4+281702072128 x^3+128970753208 x^2+32786922608 x+3544392763\right )}{1024000000 (2 x+3)^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 215, normalized size = 1.03 \begin {gather*} \frac {13251 \, \sqrt {5} {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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 (371791872 \, x^{9} + 5268182272 \, x^{8} + 40186580992 \, x^{7} + 148740043392 \, x^{6} + 304078211712 \, x^{5} + 372602220928 \, x^{4} + 281702072128 \, x^{3} + 128970753208 \, x^{2} + 32786922608 \, x + 3544392763\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{20480000000 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 614, normalized size = 2.94 \begin {gather*} \frac {13251}{10240000000} \, \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 {6784512 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 83137358592 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} + 2689605043456 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 9174489217536 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 53080570863872 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 898783135722624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 13174687008250752 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 40507172795248512 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 270169596727110016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 458790099197766656 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 1833183533173743552 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 1939024456450048032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 4903074367120921776 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3280073192617110456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 5164856211259534888 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 2082844158764403144 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 1869656136275991262 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 391066159205340747 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 153124376229353121 \, \sqrt {3} x - 9387541838830536 \, \sqrt {3} + 153124376229353121 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1024000000 \, {\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 )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 390, normalized size = 1.87 \begin {gather*} -\frac {13251 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{10240000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{51200 \left (x +\frac {3}{2}\right )^{10}}-\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{2560000 \left (x +\frac {3}{2}\right )^{8}}-\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1600000 \left (x +\frac {3}{2}\right )^{7}}-\frac {3636453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{800000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {105574503 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {3482489 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{500000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {19795101 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{2500000000}-\frac {7698831 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{10000000000}-\frac {19795101 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1250000000 \left (x +\frac {3}{2}\right )}+\frac {128093 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1600000000}-\frac {13251 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1280000000}+\frac {13251 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{10240000000}+\frac {4417 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6400000000}+\frac {13251 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40000000000}+\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10000000000}-\frac {58683 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{20000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {11989 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{6400000 \left (x +\frac {3}{2}\right )^{6}}-\frac {29 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{64000 \left (x +\frac {3}{2}\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.50, size = 579, normalized size = 2.77 \begin {gather*} \frac {316723509}{10000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} - \frac {29 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {11989 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{100000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {58683 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{625000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {3636453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {3482489 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{62500000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {105574503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {23096493}{5000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {153963369}{40000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {19795101 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500000000 \, {\left (2 \, x + 3\right )}} + \frac {384279}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2566277}{6400000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {39753}{640000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {13251}{10240000000} \, \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 {251769}{5120000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{11}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{2048 x^{11} + 33792 x^{10} + 253440 x^{9} + 1140480 x^{8} + 3421440 x^{7} + 7185024 x^{6} + 10777536 x^{5} + 11547360 x^{4} + 8660520 x^{3} + 4330260 x^{2} + 1299078 x + 177147}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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