Optimal. Leaf size=167 \[ -\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac {(5718 x+12265) \sqrt {3 x^2+5 x+2}}{512 (2 x+3)}-\frac {1875}{256} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {29047 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024 \sqrt {5}} \]
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Rubi [A] time = 0.10, 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 = {812, 810, 843, 621, 206, 724} \begin {gather*} -\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac {(5718 x+12265) \sqrt {3 x^2+5 x+2}}{512 (2 x+3)}-\frac {1875}{256} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {29047 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024 \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)^5} \, dx &=-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {5}{64} \int \frac {(-158-188 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}+\frac {\int \frac {(19556+22872 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{1024}\\ &=\frac {(12265+5718 x) \sqrt {2+5 x+3 x^2}}{512 (3+2 x)}-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {\int \frac {307624+360000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{8192}\\ &=\frac {(12265+5718 x) \sqrt {2+5 x+3 x^2}}{512 (3+2 x)}-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {5625}{256} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {29047 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1024}\\ &=\frac {(12265+5718 x) \sqrt {2+5 x+3 x^2}}{512 (3+2 x)}-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {5625}{128} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {29047}{512} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {(12265+5718 x) \sqrt {2+5 x+3 x^2}}{512 (3+2 x)}-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {1875}{256} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {29047 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 120, normalized size = 0.72 \begin {gather*} \frac {-87141 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-112500 \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 (3456 x^5-39744 x^4-533280 x^3-1672268 x^2-2059268 x-896721\right )}{(2 x+3)^4}}{15360} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.77, size = 121, normalized size = 0.72 \begin {gather*} -\frac {1875}{128} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {29047 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{512 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-3456 x^5+39744 x^4+533280 x^3+1672268 x^2+2059268 x+896721\right )}{1536 (2 x+3)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 193, normalized size = 1.16 \begin {gather*} \frac {112500 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) + 87141 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (3456 \, x^{5} - 39744 \, x^{4} - 533280 \, x^{3} - 1672268 \, x^{2} - 2059268 \, x - 896721\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{30720 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 445, normalized size = 2.66 \begin {gather*} \frac {1875}{256} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {29047}{5120} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{3072} \, {\left (\frac {\frac {10 \, {\left (\frac {195 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 904 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 18577 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 27132 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {9 \, {\left (157 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 126 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 409 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 330 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{128 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 258, normalized size = 1.54 \begin {gather*} -\frac {29047 \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 )}{5120}-\frac {1875 \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 )}{256}-\frac {\left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{75 \left (x +\frac {3}{2}\right )^{3}}-\frac {1627 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{12000 \left (x +\frac {3}{2}\right )^{2}}+\frac {1307 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{2500 \left (x +\frac {3}{2}\right )}+\frac {29047 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20000}-\frac {1387 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{2400}-\frac {461 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{320}+\frac {29047 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{9600}+\frac {29047 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{5120}-\frac {1307 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{5000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 256, normalized size = 1.53 \begin {gather*} \frac {1627}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {8 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1627 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1387}{400} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {1307}{9600} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {1307 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{1000 \, {\left (2 \, x + 3\right )}} - \frac {1383}{160} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1875}{256} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {29047}{5120} \, \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 {10607}{2560} \, \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 )}^5} \,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}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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