Optimal. Leaf size=135 \[ -\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}+\frac {3 (43 x+93) \sqrt {3 x^2+5 x+2}}{16 (2 x+3)}-\frac {343}{64} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{64 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 135, 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 = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}+\frac {3 (43 x+93) \sqrt {3 x^2+5 x+2}}{16 (2 x+3)}-\frac {343}{64} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{64 \sqrt {5}} \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 )^{3/2}}{(3+2 x)^3} \, dx &=-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {3}{32} \int \frac {(-144-172 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=\frac {3 (93+43 x) \sqrt {2+5 x+3 x^2}}{16 (3+2 x)}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}+\frac {3}{256} \int \frac {-2344-2744 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {3 (93+43 x) \sqrt {2+5 x+3 x^2}}{16 (3+2 x)}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {1029}{64} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1329}{64} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {3 (93+43 x) \sqrt {2+5 x+3 x^2}}{16 (3+2 x)}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {1029}{32} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {1329}{32} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {3 (93+43 x) \sqrt {2+5 x+3 x^2}}{16 (3+2 x)}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {343}{64} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {1329 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{64 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 110, normalized size = 0.81 \begin {gather*} \frac {1}{320} \left (-1329 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-1715 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {20 \sqrt {3 x^2+5 x+2} \left (12 x^3-142 x^2-777 x-773\right )}{(2 x+3)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 111, normalized size = 0.82 \begin {gather*} -\frac {343}{32} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {1329 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{32 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-12 x^3+142 x^2+777 x+773\right )}{16 (2 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 153, normalized size = 1.13 \begin {gather*} \frac {1715 \, \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 ) + 1329 \, \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 ) - 40 \, {\left (12 \, x^{3} - 142 \, x^{2} - 777 \, x - 773\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{640 \, {\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 = 259, normalized size = 1.92 \begin {gather*} -\frac {1}{32} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 89\right )} + \frac {1329}{320} \, \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 {343}{64} \, \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 (510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 1869 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6259 \, \sqrt {3} x + 2209 \, \sqrt {3} - 6259 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{32 \, {\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 = 179, normalized size = 1.33 \begin {gather*} -\frac {1329 \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 )}{320}-\frac {343 \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 )}{64}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}+\frac {31 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{50 \left (x +\frac {3}{2}\right )}+\frac {443 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{200}-\frac {171 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{160}+\frac {1329 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{320}-\frac {31 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{100} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 160, normalized size = 1.19 \begin {gather*} \frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {513}{80} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {343}{64} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1329}{320} \, \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 {237}{80} \, \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {31 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{20 \, {\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 )}^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}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {3 x^{3} \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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