Optimal. Leaf size=131 \[ \frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}+\frac {373 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1728}-\frac {1865 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{82944}+\frac {1865 (6 x+5) \sqrt {3 x^2+5 x+2}}{663552}-\frac {1865 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1327104 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}+\frac {373 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1728}-\frac {1865 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{82944}+\frac {1865 (6 x+5) \sqrt {3 x^2+5 x+2}}{663552}-\frac {1865 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1327104 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx &=\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {373}{48} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{3456}\\ &=-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {1865 \int \sqrt {2+5 x+3 x^2} \, dx}{55296}\\ &=\frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1327104}\\ &=\frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{663552}\\ &=\frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1327104 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 101, normalized size = 0.77 \begin {gather*} \frac {373 \left (6 \sqrt {3 x^2+5 x+2} \left (20736 x^5+86400 x^4+142128 x^3+115320 x^2+46166 x+7305\right )-5 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )\right )}{3981312}-\frac {1}{168} (14 x-71) \left (3 x^2+5 x+2\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.74, size = 89, normalized size = 0.68 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-10450944 x^7+746496 x^6+211154688 x^5+655212672 x^4+897818256 x^3+642995688 x^2+235223330 x+34777419\right )}{4644864}-\frac {1865 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{663552 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 88, normalized size = 0.67 \begin {gather*} -\frac {1}{4644864} \, {\left (10450944 \, x^{7} - 746496 \, x^{6} - 211154688 \, x^{5} - 655212672 \, x^{4} - 897818256 \, x^{3} - 642995688 \, x^{2} - 235223330 \, x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{7962624} \, \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 = 84, normalized size = 0.64 \begin {gather*} -\frac {1}{4644864} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, x - 1\right )} x - 10183\right )} x - 189587\right )} x - 2078283\right )} x - 26791487\right )} x - 117611665\right )} x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{3981312} \, \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 = 117, normalized size = 0.89 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{12}-\frac {1865 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{3981312}+\frac {71 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{168}+\frac {373 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1728}-\frac {1865 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{82944}+\frac {1865 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{663552} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 145, normalized size = 1.11 \begin {gather*} -\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {71}{168} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {373}{288} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {1865}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {1865}{13824} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {9325}{82944} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {1865}{110592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1865}{3981312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {9325}{663552} \, \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 )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,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 (- 328 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 687 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 669 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 271 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 18 x^{6} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 60 \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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