Optimal. Leaf size=160 \[ -\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}+\frac {5 (164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{192 (2 x+3)}+\frac {5 (3763-7854 x) \sqrt {3 x^2+5 x+2}}{1536}-\frac {199615 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3072 \sqrt {3}}+\frac {4295}{256} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 160, 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, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}+\frac {5 (164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{192 (2 x+3)}+\frac {5 (3763-7854 x) \sqrt {3 x^2+5 x+2}}{1536}-\frac {199615 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3072 \sqrt {3}}+\frac {4295}{256} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx &=-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {5}{64} \int \frac {(-274-328 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx\\ &=\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac {5}{512} \int \frac {(-8836-10472 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=\frac {5 (3763-7854 x) \sqrt {2+5 x+3 x^2}}{1536}+\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {5 \int \frac {545832+638768 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{24576}\\ &=\frac {5 (3763-7854 x) \sqrt {2+5 x+3 x^2}}{1536}+\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {199615 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{3072}+\frac {21475}{256} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {5 (3763-7854 x) \sqrt {2+5 x+3 x^2}}{1536}+\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {199615 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{1536}-\frac {21475}{128} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {5 (3763-7854 x) \sqrt {2+5 x+3 x^2}}{1536}+\frac {5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac {(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {199615 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{3072 \sqrt {3}}+\frac {4295}{256} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 120, normalized size = 0.75 \begin {gather*} \frac {-154620 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-199615 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {6 \sqrt {3 x^2+5 x+2} \left (1728 x^5-8544 x^4-14456 x^3-57292 x^2-290742 x-295719\right )}{(2 x+3)^2}}{9216} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.75, size = 121, normalized size = 0.76 \begin {gather*} -\frac {199615 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{1536 \sqrt {3}}+\frac {4295}{128} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-1728 x^5+8544 x^4+14456 x^3+57292 x^2+290742 x+295719\right )}{1536 (2 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 163, normalized size = 1.02 \begin {gather*} \frac {199615 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\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 ) + 154620 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\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 ) - 12 \, {\left (1728 \, x^{5} - 8544 \, x^{4} - 14456 \, x^{3} - 57292 \, x^{2} - 290742 \, x - 295719\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{18432 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 269, normalized size = 1.68 \begin {gather*} -\frac {1}{1536} \, {\left (2 \, {\left (12 \, {\left (18 \, x - 143\right )} x + 2855\right )} x - 23731\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {4295}{256} \, \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 {199615}{9216} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {5 \, {\left (4214 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 15793 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 53551 \, \sqrt {3} x + 19053 \, \sqrt {3} - 53551 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{128 \, {\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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 216, normalized size = 1.35 \begin {gather*} -\frac {4295 \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 )}{256}-\frac {199615 \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 )}{9216}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}+\frac {83 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{50 \left (x +\frac {3}{2}\right )}+\frac {859 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200}-\frac {109 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{64}-\frac {6545 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1536}+\frac {859 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{96}+\frac {4295 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256}-\frac {83 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{100} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 189, normalized size = 1.18 \begin {gather*} \frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {327}{32} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {83}{192} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {83 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{20 \, {\left (2 \, x + 3\right )}} - \frac {6545}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {199615}{9216} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {4295}{256} \, \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 {18815}{1536} \, \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 )}^3} \,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}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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