Optimal. Leaf size=167 \[ \frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{12800 (2 x+3)}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25600 \sqrt {5}} \]
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Rubi [A] time = 0.11, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {810, 812, 843, 621, 206, 724} \begin {gather*} \frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{12800 (2 x+3)}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25600 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx &=\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac {1}{160} \int \frac {(437+462 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {\int \frac {(-45914-53868 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{12800}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac {\int \frac {-725956-849600 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{102400}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {531}{128} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {137111 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{25600}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {531}{64} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {137111 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{12800}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25600 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 120, normalized size = 0.72 \begin {gather*} \frac {411333 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+531000 \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 (172800 x^5+4630848 x^4+21586808 x^3+41641148 x^2+37019838 x+12600183\right )}{(2 x+3)^5}}{384000} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.79, size = 121, normalized size = 0.72 \begin {gather*} \frac {177}{64} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {137111 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{12800 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-172800 x^5-4630848 x^4-21586808 x^3-41641148 x^2-37019838 x-12600183\right )}{38400 (2 x+3)^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 209, normalized size = 1.25 \begin {gather*} \frac {531000 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 ) + 411333 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (172800 \, x^{5} + 4630848 \, x^{4} + 21586808 \, x^{3} + 41641148 \, x^{2} + 37019838 \, x + 12600183\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{768000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 407, normalized size = 2.44 \begin {gather*} -\frac {137111}{128000} \, \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 {177}{128} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {9}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {27201072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 316934472 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 4873277176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 14374341276 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 80473660448 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 98380998102 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 236231795506 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 119385279741 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 103767800973 \, \sqrt {3} x + 13144069068 \, \sqrt {3} - 103767800973 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{38400 \, {\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 )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 279, normalized size = 1.67 \begin {gather*} \frac {137111 \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 )}{128000}+\frac {177 \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 )}{128}-\frac {521 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{15000 \left (x +\frac {3}{2}\right )^{3}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {9349 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{300000 \left (x +\frac {3}{2}\right )^{2}}+\frac {11491 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{125000}+\frac {6281 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{60000}-\frac {11491 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{62500 \left (x +\frac {3}{2}\right )}+\frac {4361 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{16000}-\frac {137111 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128000}-\frac {131 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {137111 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{240000}-\frac {137111 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{500000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 297, normalized size = 1.78 \begin {gather*} \frac {9349}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {131 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {521 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9349 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {6281}{10000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {11491}{240000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {11491 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25000 \, {\left (2 \, x + 3\right )}} + \frac {13083}{8000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {177}{128} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {137111}{128000} \, \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 {49891}{64000} \, \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 )}^{5/2}}{{\left (2\,x+3\right )}^6} \,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 {20 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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