Optimal. Leaf size=160 \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^4+\frac {229}{378} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {478}{315} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}}{68040}+\frac {25969 (6 x+5) \sqrt {3 x^2+5 x+2}}{15552}-\frac {25969 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{31104 \sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, 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}{21} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^4+\frac {229}{378} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {478}{315} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}}{68040}+\frac {25969 (6 x+5) \sqrt {3 x^2+5 x+2}}{15552}-\frac {25969 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{31104 \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)^4 \sqrt {2+5 x+3 x^2} \, dx &=-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{21} \int (3+2 x)^3 \left (\frac {707}{2}+229 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{378} \int (3+2 x)^2 \left (\frac {22377}{2}+8604 x\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {\int (3+2 x) \left (\frac {482121}{2}+189387 x\right ) \sqrt {2+5 x+3 x^2} \, dx}{5670}\\ &=\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}+\frac {25969 \int \sqrt {2+5 x+3 x^2} \, dx}{1296}\\ &=\frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{31104}\\ &=\frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{15552}\\ &=\frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{31104 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.51 \begin {gather*} \frac {-908915 \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 (1244160 x^6+1624320 x^5-28649088 x^4-123633360 x^3-208601544 x^2-161915450 x-47009103\right )}{3265920} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.91, size = 84, normalized size = 0.52 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-1244160 x^6-1624320 x^5+28649088 x^4+123633360 x^3+208601544 x^2+161915450 x+47009103\right )}{544320}-\frac {25969 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{15552 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 83, normalized size = 0.52 \begin {gather*} -\frac {1}{544320} \, {\left (1244160 \, x^{6} + 1624320 \, x^{5} - 28649088 \, x^{4} - 123633360 \, x^{3} - 208601544 \, x^{2} - 161915450 \, x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{186624} \, \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 = 79, normalized size = 0.49 \begin {gather*} -\frac {1}{544320} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, x + 47\right )} x - 24869\right )} x - 858565\right )} x - 8691731\right )} x - 80957725\right )} x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{93312} \, \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 = 0.06, size = 130, normalized size = 0.81 \begin {gather*} -\frac {16 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{4}}{21}+\frac {52 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{3}}{189}+\frac {5542 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x^{2}}{315}+\frac {34931 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{756}-\frac {25969 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{93312}+\frac {25969 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{15552}+\frac {2654033 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{68040} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 138, normalized size = 0.86 \begin {gather*} -\frac {16}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{4} + \frac {52}{189} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {5542}{315} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {34931}{756} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2654033}{68040} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {25969}{2592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {25969}{93312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {129845}{15552} \, \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 = 3.65, size = 170, normalized size = 1.06 \begin {gather*} \frac {5542\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{315}+\frac {52\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{189}-\frac {16\,x^4\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{21}-\frac {118159\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{27216}+\frac {118159\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{378}+\frac {2654033\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{1632960}+\frac {34931\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{756}+\frac {2654033\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{653184} \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 (- 999 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 864 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 264 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 16 x^{4} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 16 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 405 \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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