Optimal. Leaf size=128 \[ -\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{96} (361-726 x) \sqrt {3 x^2+5 x+2}+\frac {3743 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{192 \sqrt {3}}-\frac {161}{32} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 128, 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 = {812, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac {1}{96} (361-726 x) \sqrt {3 x^2+5 x+2}+\frac {3743 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{192 \sqrt {3}}-\frac {161}{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 812
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)^2} \, dx &=-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}-\frac {1}{8} \int \frac {(-202-242 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=-\frac {1}{96} (361-726 x) \sqrt {2+5 x+3 x^2}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {1}{384} \int \frac {12798+14972 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{96} (361-726 x) \sqrt {2+5 x+3 x^2}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {3743}{192} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {805}{32} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{96} (361-726 x) \sqrt {2+5 x+3 x^2}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {3743}{96} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {805}{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}{96} (361-726 x) \sqrt {2+5 x+3 x^2}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {3743 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{192 \sqrt {3}}-\frac {161}{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 = 0.07, size = 110, normalized size = 0.86 \begin {gather*} \frac {1}{576} \left (2898 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+3743 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {6 \sqrt {3 x^2+5 x+2} \left (48 x^3-364 x^2+256 x+1755\right )}{2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 111, normalized size = 0.87 \begin {gather*} \frac {3743 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{96 \sqrt {3}}-\frac {161}{16} \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 (-48 x^3+364 x^2-256 x-1755\right )}{96 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 139, normalized size = 1.09 \begin {gather*} \frac {3743 \, \sqrt {3} {\left (2 \, x + 3\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 ) + 2898 \, \sqrt {5} {\left (2 \, x + 3\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 ) - 12 \, {\left (48 \, x^{3} - 364 \, x^{2} + 256 \, x + 1755\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1152 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 481, normalized size = 3.76 \begin {gather*} -\frac {3743}{576} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {161}{32} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {65}{32} \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {4069 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 4308 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 14464 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 17388 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 12627 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 17928 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{96 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 158, normalized size = 1.23 \begin {gather*} \frac {161 \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 {3743 \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 )}{576}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{10 \left (x +\frac {3}{2}\right )}-\frac {161 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{60}+\frac {121 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{96}-\frac {161 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32}+\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}}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 134, normalized size = 1.05 \begin {gather*} -\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {121}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {3743}{576} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {161}{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 {361}{96} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{4 \, {\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 )}^2} \,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}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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