Optimal. Leaf size=146 \[ \frac {1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac {(51455-106734 x) \sqrt {3 x^2+5 x+2}}{27648}-\frac {543811 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{55296 \sqrt {3}}+\frac {325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 146, 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 = {814, 843, 621, 206, 724} \[ \frac {1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac {(51455-106734 x) \sqrt {3 x^2+5 x+2}}{27648}-\frac {543811 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{55296 \sqrt {3}}+\frac {325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
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} \, dx &=\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{144} \int \frac {(1623+1862 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac {(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {\int \frac {(-179802-213468 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{13824}\\ &=\frac {(51455-106734 x) \sqrt {2+5 x+3 x^2}}{27648}+\frac {(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {\int \frac {11153196+13051464 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{663552}\\ &=\frac {(51455-106734 x) \sqrt {2+5 x+3 x^2}}{27648}+\frac {(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {543811 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{55296}+\frac {1625}{128} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {(51455-106734 x) \sqrt {2+5 x+3 x^2}}{27648}+\frac {(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {543811 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{27648}-\frac {1625}{64} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {(51455-106734 x) \sqrt {2+5 x+3 x^2}}{27648}+\frac {(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac {1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {543811 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{55296 \sqrt {3}}+\frac {325}{128} \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 = 0.06, size = 113, normalized size = 0.77 \[ \frac {-2106000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-2719055 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (103680 x^5-376704 x^4-1311120 x^3-1624872 x^2-583490 x-580299\right )}{829440} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 129, normalized size = 0.88 \[ -\frac {1}{138240} \, {\left (103680 \, x^{5} - 376704 \, x^{4} - 1311120 \, x^{3} - 1624872 \, x^{2} - 583490 \, x - 580299\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {543811}{331776} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac {325}{256} \, \sqrt {5} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 146, normalized size = 1.00 \[ -\frac {1}{138240} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 109\right )} x - 3035\right )} x - 67703\right )} x - 291745\right )} x - 580299\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {325}{128} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {543811}{165888} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 239, normalized size = 1.64 \[ -\frac {325 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{128}-\frac {7553 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{2304}+\frac {5 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{165888}-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{72}+\frac {5 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{3456}-\frac {5 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{27648}+\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20}-\frac {13 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{48}-\frac {247 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{384}+\frac {65 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{48}+\frac {325 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 157, normalized size = 1.08 \[ -\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {209}{360} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {931}{576} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {25}{3456} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {17789}{4608} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {543811}{165888} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {325}{128} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {51455}{27648} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{2\,x+3} \,d x \]
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 \[ - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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