Optimal. Leaf size=141 \[ \frac{1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3+\frac{8}{63} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^2+\frac{13 (50 x+29) \left (3 x^2-x+2\right )^{5/2}}{2520}+\frac{91 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{3456}+\frac{2093 (1-6 x) \sqrt{3 x^2-x+2}}{27648}+\frac{48139 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{55296 \sqrt{3}} \]
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Rubi [A] time = 0.12095, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{1}{12} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^3+\frac{8}{63} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^2+\frac{13 (50 x+29) \left (3 x^2-x+2\right )^{5/2}}{2520}+\frac{91 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{3456}+\frac{2093 (1-6 x) \sqrt{3 x^2-x+2}}{27648}+\frac{48139 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{55296 \sqrt{3}} \]
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 )^{3/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{96} \int (1+2 x)^2 (20+256 x) \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{\int (1+2 x) (-988+4680 x) \left (2-x+3 x^2\right )^{3/2} \, dx}{2016}\\ &=\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}-\frac{91}{144} \int \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{91 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{3456}+\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}-\frac{2093 \int \sqrt{2-x+3 x^2} \, dx}{2304}\\ &=\frac{2093 (1-6 x) \sqrt{2-x+3 x^2}}{27648}+\frac{91 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{3456}+\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}-\frac{48139 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{55296}\\ &=\frac{2093 (1-6 x) \sqrt{2-x+3 x^2}}{27648}+\frac{91 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{3456}+\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}-\frac{\left (2093 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{55296}\\ &=\frac{2093 (1-6 x) \sqrt{2-x+3 x^2}}{27648}+\frac{91 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{3456}+\frac{8}{63} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{12} (1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}+\frac{13 (29+50 x) \left (2-x+3 x^2\right )^{5/2}}{2520}+\frac{48139 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{55296 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0433892, size = 75, normalized size = 0.53 \[ \frac{6 \sqrt{3 x^2-x+2} \left (5806080 x^7+9262080 x^6+10656000 x^5+12173952 x^4+10119792 x^3+5694024 x^2+2735918 x+1517367\right )-1684865 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{5806080} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 117, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{3}}{3} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{95\,{x}^{2}}{63} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{319\,x}{252} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{-2093+12558\,x}{27648}\sqrt{3\,{x}^{2}-x+2}}-{\frac{48139\,\sqrt{3}}{165888}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{-91+546\,x}{3456} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{907}{2520} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50437, size = 186, normalized size = 1.32 \begin{align*} \frac{2}{3} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{95}{63} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{319}{252} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{907}{2520} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} - \frac{91}{576} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{91}{3456} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{2093}{4608} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{48139}{165888} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{2093}{27648} \, \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 = 1.62997, size = 308, normalized size = 2.18 \begin{align*} \frac{1}{967680} \,{\left (5806080 \, x^{7} + 9262080 \, x^{6} + 10656000 \, x^{5} + 12173952 \, x^{4} + 10119792 \, x^{3} + 5694024 \, x^{2} + 2735918 \, x + 1517367\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{48139}{331776} \, \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{3}{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 = 1.15427, size = 112, normalized size = 0.79 \begin{align*} \frac{1}{967680} \,{\left (2 \,{\left (12 \,{\left (2 \,{\left (8 \,{\left (30 \,{\left (12 \,{\left (42 \, x + 67\right )} x + 925\right )} x + 31703\right )} x + 210829\right )} x + 237251\right )} x + 1367959\right )} x + 1517367\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{48139}{165888} \, \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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