Optimal. Leaf size=62 \[ \frac {1}{6} \sqrt {3 x^2+5 x+2} (19-2 x)+\frac {31 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {779, 621, 206} \begin {gather*} \frac {1}{6} \sqrt {3 x^2+5 x+2} (19-2 x)+\frac {31 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)}{\sqrt {2+5 x+3 x^2}} \, dx &=\frac {1}{6} (19-2 x) \sqrt {2+5 x+3 x^2}+\frac {31}{4} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{6} (19-2 x) \sqrt {2+5 x+3 x^2}+\frac {31}{2} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {1}{6} (19-2 x) \sqrt {2+5 x+3 x^2}+\frac {31 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.92 \begin {gather*} \frac {1}{12} \left (31 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-2 (2 x-19) \sqrt {3 x^2+5 x+2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 59, normalized size = 0.95 \begin {gather*} \frac {1}{6} \sqrt {3 x^2+5 x+2} (19-2 x)+\frac {31 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 58, normalized size = 0.94 \begin {gather*} -\frac {1}{6} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x - 19\right )} + \frac {31}{24} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 54, normalized size = 0.87 \begin {gather*} -\frac {1}{6} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x - 19\right )} - \frac {31}{12} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {3 x^{2}+5 x +2}\, x}{3}+\frac {31 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{12}+\frac {19 \sqrt {3 x^{2}+5 x +2}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 58, normalized size = 0.94 \begin {gather*} -\frac {1}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {31}{12} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {19}{6} \, \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 = -0.02 \begin {gather*} -\int \frac {\left (2\,x+3\right )\,\left (x-5\right )}{\sqrt {3\,x^2+5\,x+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 {7 x}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {2 x^{2}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {15}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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