Optimal. Leaf size=158 \[ -\frac {1}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac {67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac {(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac {12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac {12277 (6 x+5) \sqrt {3 x^2+5 x+2}}{165888}+\frac {12277 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{331776 \sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 158, 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}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac {67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac {(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac {12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac {12277 (6 x+5) \sqrt {3 x^2+5 x+2}}{165888}+\frac {12277 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{331776 \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 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{24} \int (3+2 x)^2 \left (\frac {819}{2}+268 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{504} \int (3+2 x) \left (\frac {27209}{2}+9963 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac {12277}{864} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}-\frac {12277 \int \sqrt {2+5 x+3 x^2} \, dx}{13824}\\ &=-\frac {12277 (5+6 x) \sqrt {2+5 x+3 x^2}}{165888}+\frac {12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac {12277 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{331776}\\ &=-\frac {12277 (5+6 x) \sqrt {2+5 x+3 x^2}}{165888}+\frac {12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac {12277 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{165888}\\ &=-\frac {12277 (5+6 x) \sqrt {2+5 x+3 x^2}}{165888}+\frac {12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac {12277 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{331776 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 87, normalized size = 0.55 \begin {gather*} \frac {429695 \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 (17418240 x^7+25297920 x^6-368236800 x^5-1650151296 x^4-2993047920 x^3-2762417688 x^2-1276112350 x-233137461\right )}{34836480} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.79, size = 89, normalized size = 0.56 \begin {gather*} \frac {12277 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{165888 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-17418240 x^7-25297920 x^6+368236800 x^5+1650151296 x^4+2993047920 x^3+2762417688 x^2+1276112350 x+233137461\right )}{5806080} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 88, normalized size = 0.56 \begin {gather*} -\frac {1}{5806080} \, {\left (17418240 \, x^{7} + 25297920 \, x^{6} - 368236800 \, x^{5} - 1650151296 \, x^{4} - 2993047920 \, x^{3} - 2762417688 \, x^{2} - 1276112350 \, x - 233137461\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {12277}{1990656} \, \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 = 0.21, size = 84, normalized size = 0.53 \begin {gather*} -\frac {1}{5806080} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (12 \, {\left (42 \, x + 61\right )} x - 10655\right )} x - 1432423\right )} x - 20785055\right )} x - 115100737\right )} x - 638056175\right )} x - 233137461\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {12277}{995328} \, \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 = 132, normalized size = 0.84 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{3}}{3}+\frac {79 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{2}}{126}+\frac {1063 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{168}+\frac {12277 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{995328}-\frac {12277 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{165888}+\frac {12277 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{20736}+\frac {130801 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 150, normalized size = 0.95 \begin {gather*} -\frac {1}{3} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {79}{126} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {1063}{168} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {130801}{15120} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {12277}{3456} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {61385}{20736} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {12277}{27648} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {12277}{995328} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {61385}{165888} \, \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 {\left (2\,x+3\right )}^3\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,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 (- 1161 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1872 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1367 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 382 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 28 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 24 x^{6} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 270 \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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