Optimal. Leaf size=165 \[ -\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac {5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac {5 (343 x+736) \sqrt {3 x^2+5 x+2}}{64 (2 x+3)}+\frac {13505 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}-\frac {3487}{256} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 165, 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 = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac {5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac {5 (343 x+736) \sqrt {3 x^2+5 x+2}}{64 (2 x+3)}+\frac {13505 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}-\frac {3487}{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 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx &=-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac {5}{72} \int \frac {(-216-258 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx\\ &=\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {5}{768} \int \frac {(-7032-8232 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac {5 \int \frac {-110784-129648 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{6144}\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505}{256} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {17435}{256} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505}{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 {17435}{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 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{256 \sqrt {3}}-\frac {3487}{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 = 0.10, size = 120, normalized size = 0.73 \begin {gather*} \frac {1}{768} \left (10461 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+13505 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {4 \sqrt {3 x^2+5 x+2} \left (288 x^5-1896 x^4+1944 x^3+64332 x^2+143533 x+89224\right )}{(2 x+3)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.75, size = 121, normalized size = 0.73 \begin {gather*} \frac {13505 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{128 \sqrt {3}}-\frac {3487}{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 (-288 x^5+1896 x^4-1944 x^3-64332 x^2-143533 x-89224\right )}{192 (2 x+3)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 179, normalized size = 1.08 \begin {gather*} \frac {13505 \, \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 ) + 10461 \, \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 ) - 8 \, {\left (288 \, x^{5} - 1896 \, x^{4} + 1944 \, x^{3} + 64332 \, x^{2} + 143533 \, x + 89224\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1536 \, {\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 = 315, normalized size = 1.91 \begin {gather*} -\frac {1}{128} \, {\left (2 \, {\left (12 \, x - 133\right )} x + 1197\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {3487}{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 {13505}{768} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {203604 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1334970 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 10053790 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 12051375 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 20819415 \, \sqrt {3} x + 4639299 \, \sqrt {3} - 20819415 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{384 \, {\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 = 237, normalized size = 1.44 \begin {gather*} \frac {3487 \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 {13505 \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 )}{768}+\frac {67 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {197 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{125 \left (x +\frac {3}{2}\right )}-\frac {3487 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000}+\frac {329 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{240}+\frac {443 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{128}-\frac {3487 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{480}-\frac {3487 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256}+\frac {197 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{250}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{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 = 220, normalized size = 1.33 \begin {gather*} -\frac {67}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {67 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {329}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {197}{480} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {197 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50 \, {\left (2 \, x + 3\right )}} + \frac {1329}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {13505}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {3487}{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 {159}{16} \, \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 )}^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 {20 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {96 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 {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {9 x^{5} \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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