Optimal. Leaf size=164 \[ \frac{1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}+\frac{37}{405} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{(3430 x+2731) \left (3 x^2-x+2\right )^{7/2}}{17010}-\frac{293 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{58320}-\frac{6739 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}-\frac{154997 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}} \]
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Rubi [A] time = 0.132842, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1653, 832, 779, 612, 619, 215} \[ \frac{1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}+\frac{37}{405} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{(3430 x+2731) \left (3 x^2-x+2\right )^{7/2}}{17010}-\frac{293 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{58320}-\frac{6739 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}-\frac{154997 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 832
Rule 779
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{120} \int (1+2 x)^2 (52+296 x) \left (2-x+3 x^2\right )^{5/2} \, dx\\ &=\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{\int (1+2 x) (72+7840 x) \left (2-x+3 x^2\right )^{5/2} \, dx}{3240}\\ &=\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{293 \int \left (2-x+3 x^2\right )^{5/2} \, dx}{1620}\\ &=-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{6739 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{154997 \int \sqrt{2-x+3 x^2} \, dx}{373248}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{3564931 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{8957952}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{\left (154997 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{8957952}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0550144, size = 85, normalized size = 0.52 \[ \frac{6 \sqrt{3 x^2-x+2} \left (2257403904 x^9+2675441664 x^8+4427716608 x^7+5671627776 x^6+4996802304 x^5+4171579776 x^4+3096104976 x^3+1693765752 x^2+692659234 x+387182961\right )+124772585 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{940584960} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 136, normalized size = 0.8 \begin{align*}{\frac{5419}{17010} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{x}^{3}}{15} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{472\,{x}^{2}}{405} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{235\,x}{243} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{-154997+929982\,x}{4478976}\sqrt{3\,{x}^{2}-x+2}}+{\frac{3564931\,\sqrt{3}}{26873856}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{-293+1758\,x}{58320} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-6739+40434\,x}{559872} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48004, size = 225, normalized size = 1.37 \begin{align*} \frac{8}{15} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{472}{405} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{235}{243} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x + \frac{5419}{17010} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} + \frac{293}{9720} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x - \frac{293}{58320} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{6739}{93312} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{6739}{559872} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{154997}{746496} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{3564931}{26873856} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{154997}{4478976} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32793, size = 390, normalized size = 2.38 \begin{align*} \frac{1}{156764160} \,{\left (2257403904 \, x^{9} + 2675441664 \, x^{8} + 4427716608 \, x^{7} + 5671627776 \, x^{6} + 4996802304 \, x^{5} + 4171579776 \, x^{4} + 3096104976 \, x^{3} + 1693765752 \, x^{2} + 692659234 \, x + 387182961\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{3564931}{53747712} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x + 1\right )^{2} \left (3 x^{2} - x + 2\right )^{\frac{5}{2}} \left (4 x^{2} + 3 x + 1\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19182, size = 126, normalized size = 0.77 \begin{align*} \frac{1}{156764160} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (24 \,{\left (27 \, x + 32\right )} x + 1271\right )} x + 22793\right )} x + 722917\right )} x + 3621163\right )} x + 21500729\right )} x + 70573573\right )} x + 346329617\right )} x + 387182961\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{3564931}{26873856} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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