Optimal. Leaf size=110 \[ -\frac {1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac {2267 (6 x+5) \sqrt {3 x^2+5 x+2}}{2592}-\frac {2267 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5184 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac {2267 (6 x+5) \sqrt {3 x^2+5 x+2}}{2592}-\frac {2267 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5184 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \sqrt {2+5 x+3 x^2} \, dx &=-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{15} \int (3+2 x) \left (\frac {511}{2}+167 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}+\frac {2267}{216} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{5184}\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{2592}\\ &=\frac {2267 (5+6 x) \sqrt {2+5 x+3 x^2}}{2592}-\frac {1}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(7969+3006 x) \left (2+5 x+3 x^2\right )^{3/2}}{1620}-\frac {2267 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{5184 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.65 \begin {gather*} \frac {-11335 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (10368 x^4-23760 x^3-229416 x^2-375250 x-168627\right )}{77760} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 74, normalized size = 0.67 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-10368 x^4+23760 x^3+229416 x^2+375250 x+168627\right )}{12960}-\frac {2267 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2592 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 73, normalized size = 0.66 \begin {gather*} -\frac {1}{12960} \, {\left (10368 \, x^{4} - 23760 \, x^{3} - 229416 \, x^{2} - 375250 \, x - 168627\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2267}{31104} \, \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 = 0.19, size = 69, normalized size = 0.63 \begin {gather*} -\frac {1}{12960} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (24 \, x - 55\right )} x - 9559\right )} x - 187625\right )} x - 168627\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2267}{15552} \, \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 = 0.05, size = 96, normalized size = 0.87 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {19 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{18}-\frac {2267 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{15552}+\frac {6997 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{1620}+\frac {2267 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{2592} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 104, normalized size = 0.95 \begin {gather*} -\frac {4}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {19}{18} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {6997}{1620} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {2267}{432} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {2267}{15552} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {11335}{2592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 136, normalized size = 1.24 \begin {gather*} \frac {386\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{9}-\frac {193\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{324}-\frac {4\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{15}+\frac {6997\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{38880}+\frac {19\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{18}+\frac {6997\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{15552} \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 (- 51 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 8 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 4 x^{3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 45 \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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