Optimal. Leaf size=124 \[ \frac{2}{15} \left (3 x^2-x+2\right )^{5/2}+\frac{1}{144} (30 x+7) \left (3 x^2-x+2\right )^{3/2}+\frac{(402 x+869) \sqrt{3 x^2-x+2}}{1152}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}} \]
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Rubi [A] time = 0.144169, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1653, 814, 843, 619, 215, 724, 206} \[ \frac{2}{15} \left (3 x^2-x+2\right )^{5/2}+\frac{1}{144} (30 x+7) \left (3 x^2-x+2\right )^{3/2}+\frac{(402 x+869) \sqrt{3 x^2-x+2}}{1152}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 814
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{1}{60} \int \frac{(80+100 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx\\ &=\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{\int \frac{(-13380-8040 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{5760}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{\int \frac{1195800-528720 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{276480}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{2203 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{2304}+\frac{169}{32} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}-\frac{169}{16} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{2203 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{2304 \sqrt{69}}\\ &=\frac{(869+402 x) \sqrt{2-x+3 x^2}}{1152}+\frac{1}{144} (7+30 x) \left (2-x+3 x^2\right )^{3/2}+\frac{2}{15} \left (2-x+3 x^2\right )^{5/2}+\frac{2203 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{2304 \sqrt{3}}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0577476, size = 96, normalized size = 0.77 \[ \frac{6 \sqrt{3 x^2-x+2} \left (6912 x^4-1008 x^3+9624 x^2+1058 x+7977\right )-14040 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-11015 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{34560} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 151, normalized size = 1.2 \begin{align*}{\frac{2}{15} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-5+30\,x}{144} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-115+690\,x}{1152}\sqrt{3\,{x}^{2}-x+2}}-{\frac{2203\,\sqrt{3}}{6912}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1}{12} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{-1+6\,x}{24}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}+{\frac{13}{32}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{13\,\sqrt{13}}{32}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48406, size = 169, normalized size = 1.36 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{5}{24} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{7}{144} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{67}{192} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{2203}{6912} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{13}{32} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{869}{1152} \, \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 = 1.6081, size = 367, normalized size = 2.96 \begin{align*} \frac{1}{5760} \,{\left (6912 \, x^{4} - 1008 \, x^{3} + 9624 \, x^{2} + 1058 \, x + 7977\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{2203}{13824} \, \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 ) + \frac{13}{64} \, \sqrt{13} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\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 \frac{\left (3 x^{2} - x + 2\right )^{\frac{3}{2}} \left (4 x^{2} + 3 x + 1\right )}{2 x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28475, size = 184, normalized size = 1.48 \begin{align*} \frac{1}{5760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (48 \, x - 7\right )} x + 401\right )} x + 529\right )} x + 7977\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{2203}{6912} \, \sqrt{3} \log \left (-6 \, \sqrt{3} x + \sqrt{3} + 6 \, \sqrt{3 \, x^{2} - x + 2}\right ) + \frac{13}{32} \, \sqrt{13} \log \left (-\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{13} - 2 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} - x + 2} \right |}}{2 \,{\left (2 \, \sqrt{3} x - \sqrt{13} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} - x + 2}\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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