Optimal. Leaf size=135 \[ -\frac {1}{18} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {11}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}}{3240}+\frac {6221 (6 x+5) \sqrt {3 x^2+5 x+2}}{5184}-\frac {6221 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{10368 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 135, 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 = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{18} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {11}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}}{3240}+\frac {6221 (6 x+5) \sqrt {3 x^2+5 x+2}}{5184}-\frac {6221 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{10368 \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)^3 \sqrt {2+5 x+3 x^2} \, dx &=-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{18} \int (3+2 x)^2 \left (\frac {609}{2}+198 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{270} \int (3+2 x) \left (\frac {15327}{2}+5769 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}+\frac {6221}{432} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{10368}\\ &=\frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{5184}\\ &=\frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10368 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.57 \begin {gather*} \frac {-31105 \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 (34560 x^5-14976 x^4-825840 x^3-2317848 x^2-2432350 x-859701\right )}{155520} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 79, normalized size = 0.59 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-34560 x^5+14976 x^4+825840 x^3+2317848 x^2+2432350 x+859701\right )}{25920}-\frac {6221 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{5184 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 78, normalized size = 0.58 \begin {gather*} -\frac {1}{25920} \, {\left (34560 \, x^{5} - 14976 \, x^{4} - 825840 \, x^{3} - 2317848 \, x^{2} - 2432350 \, x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{62208} \, \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 = 0.20, size = 74, normalized size = 0.55 \begin {gather*} -\frac {1}{25920} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, x - 13\right )} x - 5735\right )} x - 96577\right )} x - 1216175\right )} x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{31104} \, \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 = 113, normalized size = 0.84 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{3}}{9}+\frac {14 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {337 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{36}-\frac {6221 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{31104}+\frac {44011 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{3240}+\frac {6221 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{5184} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 121, normalized size = 0.90 \begin {gather*} -\frac {4}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {14}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {337}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {44011}{3240} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {6221}{864} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {6221}{31104} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {31105}{5184} \, \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 = 3.47, size = 153, normalized size = 1.13 \begin {gather*} \frac {14\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{15}-\frac {4\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{9}-\frac {2093\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{1296}+\frac {2093\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{18}+\frac {44011\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{77760}+\frac {337\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{36}+\frac {44011\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{31104} \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 (- 243 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 126 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 8 x^{4} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 135 \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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