Optimal. Leaf size=100 \[ \frac {1}{24} \sqrt {3 x^2+5 x+2} (73-6 x)-\frac {311 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, 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}{24} \sqrt {3 x^2+5 x+2} (73-6 x)-\frac {311 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \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) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx &=\frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{48} \int \frac {543+622 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311}{48} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {65}{8} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311}{24} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {65}{4} \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}{24} (73-6 x) \sqrt {2+5 x+3 x^2}-\frac {311 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{48 \sqrt {3}}+\frac {13}{8} \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 = 93, normalized size = 0.93 \begin {gather*} \frac {1}{144} \left (-6 \sqrt {3 x^2+5 x+2} (6 x-73)-234 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-311 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 94, normalized size = 0.94 \begin {gather*} \frac {1}{24} \sqrt {3 x^2+5 x+2} (73-6 x)-\frac {311 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{24 \sqrt {3}}+\frac {13}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 109, normalized size = 1.09 \begin {gather*} -\frac {1}{24} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 73\right )} + \frac {311}{288} \, \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 {13}{16} \, \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 = 0.29, size = 126, normalized size = 1.26 \begin {gather*} -\frac {1}{24} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 73\right )} + \frac {13}{8} \, \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 {311}{144} \, \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 = 127, normalized size = 1.27 \begin {gather*} -\frac {13 \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 )}{8}-\frac {13 \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 )}{6}+\frac {\sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{144}-\frac {\left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{24}+\frac {13 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 99, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {311}{144} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {13}{8} \, \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 {73}{24} \, \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 )\,\sqrt {3\,x^2+5\,x+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 {5 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {x \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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