Optimal. Leaf size=137 \[ \frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{160 (2 x+3)}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{160 (2 x+3)}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320 \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 )^{3/2}}{(3+2 x)^4} \, dx &=\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}-\frac {1}{80} \int \frac {(361+402 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {1}{640} \int \frac {5234+6120 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {153}{32} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {1973}{320} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {153}{16} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {1973}{160} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{320 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 110, normalized size = 0.80 \begin {gather*} \frac {5919 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+7650 \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 (720 x^3+13176 x^2+30878 x+19751\right )}{(2 x+3)^3}}{4800} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 111, normalized size = 0.81 \begin {gather*} \frac {51}{16} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {1973 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{160 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-720 x^3-13176 x^2-30878 x-19751\right )}{480 (2 x+3)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 169, normalized size = 1.23 \begin {gather*} \frac {7650 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) + 5919 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 (720 \, x^{3} + 13176 \, x^{2} + 30878 \, x + 19751\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9600 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 305, normalized size = 2.23 \begin {gather*} -\frac {1973}{1600} \, \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 {51}{32} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {3}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {62484 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 390510 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2835190 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3307455 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 5598195 \, \sqrt {3} x + 1227924 \, \sqrt {3} - 5598195 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{480 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 200, normalized size = 1.46 \begin {gather*} \frac {1973 \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 )}{1600}+\frac {51 \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 )}{32}-\frac {37 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {158 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{375 \left (x +\frac {3}{2}\right )}-\frac {1973 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3000}+\frac {121 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{400}-\frac {1973 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1600}+\frac {79 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{375}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 191, normalized size = 1.39 \begin {gather*} \frac {37}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {37 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {363}{200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {51}{32} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1973}{1600} \, \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 {763}{800} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {79 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{75 \, {\left (2 \, x + 3\right )}} \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 )}^{3/2}}{{\left (2\,x+3\right )}^4} \,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 {10 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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