Optimal. Leaf size=123 \[ \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{128} (175-414 x) \sqrt {3 x^2+5 x+2}-\frac {2011 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{128} (175-414 x) \sqrt {3 x^2+5 x+2}-\frac {2011 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \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 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx &=\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{96} \int \frac {(1083+1242 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac {\int \frac {-61794-72396 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{4608}\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011}{256} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {325}{32} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011}{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 {325}{16} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \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 = 0.04, size = 103, normalized size = 0.84 \begin {gather*} \frac {1}{768} \left (-1560 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-2011 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-2 \sqrt {3 x^2+5 x+2} \left (144 x^3-888 x^2-542 x-1277\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 104, normalized size = 0.85 \begin {gather*} -\frac {2011 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{128 \sqrt {3}}+\frac {65}{16} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {1}{384} \sqrt {3 x^2+5 x+2} \left (-144 x^3+888 x^2+542 x+1277\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 119, normalized size = 0.97 \begin {gather*} -\frac {1}{384} \, {\left (144 \, x^{3} - 888 \, x^{2} - 542 \, x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2011}{1536} \, \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 {65}{64} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 136, normalized size = 1.11 \begin {gather*} -\frac {1}{384} \, {\left (2 \, {\left (12 \, {\left (6 \, x - 37\right )} x - 271\right )} x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {65}{32} \, \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 {2011}{768} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 183, normalized size = 1.49 \begin {gather*} -\frac {65 \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 )}{32}-\frac {377 \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 )}{144}-\frac {\sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{2304}-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{48}+\frac {\left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{384}+\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {13 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{24}+\frac {65 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 128, normalized size = 1.04 \begin {gather*} -\frac {1}{8} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {47}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {207}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {2011}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {65}{32} \, \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 {175}{128} \, \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 )}^{3/2}}{2\,x+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 {10 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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