Optimal. Leaf size=197 \[ \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac {3 (434104 x+559841) \sqrt {3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{8192000 \sqrt {5}} \]
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Rubi [A] time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {810, 843, 621, 206, 724} \begin {gather*} \frac {(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac {(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac {(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac {3 (434104 x+559841) \sqrt {3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{8192000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 843
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\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx &=\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {1}{320} \int \frac {(291+240 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac {\int \frac {(-36690-43200 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{76800}\\ &=\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {\int \frac {(4488660+5184000 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{12288000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac {\int \frac {-265774680-311040000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{983040000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {81}{512} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1673211 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{8192000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {81}{256} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {1673211 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{4096000}\\ &=\frac {3 (559841+434104 x) \sqrt {2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac {(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac {(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac {(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac {27}{512} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{8192000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 130, normalized size = 0.66 \begin {gather*} \frac {-11712477 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-15120000 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+\frac {10 \sqrt {3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{(2 x+3)^8}}{286720000} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.99, size = 131, normalized size = 0.66 \begin {gather*} -\frac {27}{256} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {1673211 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{4096000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{28672000 (2 x+3)^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 263, normalized size = 1.34 \begin {gather*} \frac {15120000 \, \sqrt {3} {\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 )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 11712477 \, \sqrt {5} {\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 )} \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 (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{573440000 \, {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 546, normalized size = 2.77 \begin {gather*} \frac {1673211}{40960000} \, \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 {27}{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 {25982914944 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 475461282240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 12329944383680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 66497191380480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 747738478510240 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 2056338758898032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 12823219634258640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 20470141041874560 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 75774797457107080 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 72179382871515780 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 157788604924552196 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 86325470670757920 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 102935771527447390 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 28057073003987265 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 14067886443441495 \, \sqrt {3} x + 1086949713645432 \, \sqrt {3} - 14067886443441495 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{28672000 \, {\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 )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 379, normalized size = 1.92 \begin {gather*} -\frac {1673211 \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 )}{40960000}-\frac {27 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{512}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}-\frac {81 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{44800 \left (x +\frac {3}{2}\right )^{7}}-\frac {158331 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{22400000 \left (x +\frac {3}{2}\right )^{4}}-\frac {664383 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{40000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {150503 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{14000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {767427 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{70000000}-\frac {135591 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40000000}-\frac {767427 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{35000000 \left (x +\frac {3}{2}\right )}-\frac {25627 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6400000}-\frac {53211 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{5120000}+\frac {1673211 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{40960000}+\frac {557737 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{25600000}+\frac {1673211 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{160000000}+\frac {1673211 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{280000000}-\frac {363 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{80000 \left (x +\frac {3}{2}\right )^{5}}-\frac {523 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{179200 \left (x +\frac {3}{2}\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.64, size = 479, normalized size = 2.43 \begin {gather*} \frac {1993149}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{40 \, {\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 {81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{350 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {523 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2800 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {158331 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1400000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {150503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {664383 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {406773}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {1038609}{160000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {767427 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{14000000 \, {\left (2 \, x + 3\right )}} - \frac {76881}{3200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {45197}{25600000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {159633}{2560000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {27}{512} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1673211}{40960000} \, \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 {608991}{20480000} \, \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 = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^9} \,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}}{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 )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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