Optimal. Leaf size=147 \[ \frac{2}{21} \left (3 x^2-x+2\right )^{7/2}+\frac{(150 x+29) \left (3 x^2-x+2\right )^{5/2}}{1080}+\frac{(2154 x+2449) \left (3 x^2-x+2\right )^{3/2}}{10368}+\frac{(221999-17850 x) \sqrt{3 x^2-x+2}}{82944}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}} \]
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Rubi [A] time = 0.159525, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2}{21} \left (3 x^2-x+2\right )^{7/2}+\frac{(150 x+29) \left (3 x^2-x+2\right )^{5/2}}{1080}+\frac{(2154 x+2449) \left (3 x^2-x+2\right )^{3/2}}{10368}+\frac{(221999-17850 x) \sqrt{3 x^2-x+2}}{82944}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}} \]
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 )^{5/2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{1}{84} \int \frac{(112+140 x) \left (2-x+3 x^2\right )^{5/2}}{1+2 x} \, dx\\ &=\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{\int \frac{(-29708-20104 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx}{12096}\\ &=\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{\int \frac{(5632872-999600 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{1161216}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{\int \frac{-639337776+634718112 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{55738368}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{944521 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{165888}+\frac{2197}{128} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}-\frac{2197}{64} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{944521 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{165888 \sqrt{69}}\\ &=\frac{(221999-17850 x) \sqrt{2-x+3 x^2}}{82944}+\frac{(2449+2154 x) \left (2-x+3 x^2\right )^{3/2}}{10368}+\frac{(29+150 x) \left (2-x+3 x^2\right )^{5/2}}{1080}+\frac{2}{21} \left (2-x+3 x^2\right )^{7/2}+\frac{944521 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{165888 \sqrt{3}}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0735996, size = 106, normalized size = 0.72 \[ \frac{6 \sqrt{3 x^2-x+2} \left (7464960 x^6-3836160 x^5+15700608 x^4-3646512 x^3+12466776 x^2-2120998 x+11665053\right )-22997520 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-33058235 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{17418240} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 207, normalized size = 1.4 \begin{align*}{\frac{2}{21} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{-5+30\,x}{216} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-575+3450\,x}{10368} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-13225+79350\,x}{82944}\sqrt{3\,{x}^{2}-x+2}}-{\frac{944521\,\sqrt{3}}{497664}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1}{20} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{-1+6\,x}{48} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{-25+150\,x}{128}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}+{\frac{13}{48} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{169}{128}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{169\,\sqrt{13}}{128}{\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 = 1.51275, size = 208, normalized size = 1.41 \begin{align*} \frac{2}{21} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} + \frac{5}{36} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{29}{1080} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{359}{1728} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{2449}{10368} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{2975}{13824} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{944521}{497664} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{169}{128} \, \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{221999}{82944} \, \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 = 1.37156, size = 440, normalized size = 2.99 \begin{align*} \frac{1}{2903040} \,{\left (7464960 \, x^{6} - 3836160 \, x^{5} + 15700608 \, x^{4} - 3646512 \, x^{3} + 12466776 \, x^{2} - 2120998 \, x + 11665053\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{944521}{995328} \, \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{169}{256} \, \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{5}{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 = 1.34667, size = 197, normalized size = 1.34 \begin{align*} \frac{1}{2903040} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \,{\left (72 \, x - 37\right )} x + 4543\right )} x - 8441\right )} x + 519449\right )} x - 1060499\right )} x + 11665053\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{944521}{497664} \, \sqrt{3} \log \left (-6 \, \sqrt{3} x + \sqrt{3} + 6 \, \sqrt{3 \, x^{2} - x + 2}\right ) + \frac{169}{128} \, \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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