Optimal. Leaf size=154 \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{13 (2 x+1)}-\frac{11 (37-60 x) \left (3 x^2-x+2\right )^{5/2}}{2340}-\frac{11}{864} (67-78 x) \left (3 x^2-x+2\right )^{3/2}-\frac{11 (4727-3090 x) \sqrt{3 x^2-x+2}}{6912}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}} \]
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Rubi [A] time = 0.163071, antiderivative size = 154, 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 = {1650, 814, 843, 619, 215, 724, 206} \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{13 (2 x+1)}-\frac{11 (37-60 x) \left (3 x^2-x+2\right )^{5/2}}{2340}-\frac{11}{864} (67-78 x) \left (3 x^2-x+2\right )^{3/2}-\frac{11 (4727-3090 x) \sqrt{3 x^2-x+2}}{6912}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1650
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)^2} \, dx &=-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{1}{13} \int \frac{\left (-\frac{11}{2}-44 x\right ) \left (2-x+3 x^2\right )^{5/2}}{1+2 x} \, dx\\ &=-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{\int \frac{(-286+14872 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx}{1872}\\ &=-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{\int \frac{(641784-3534960 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{179712}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{\int \frac{-178896432+393897504 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{8626176}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{315623 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{13824}-\frac{5577}{128} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{5577}{64} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )+\frac{315623 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{13824 \sqrt{69}}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}}+\frac{429}{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 = 0.115087, size = 113, normalized size = 0.73 \[ \frac{\frac{6 \sqrt{3 x^2-x+2} \left (103680 x^6-65664 x^5+251424 x^4-115680 x^3+310660 x^2-322972 x-364257\right )}{2 x+1}+694980 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+1578115 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{207360} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 235, normalized size = 1.5 \begin{align*}{\frac{-1+6\,x}{36} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-115+690\,x}{1728} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-2645+15870\,x}{13824}\sqrt{3\,{x}^{2}-x+2}}+{\frac{315623\,\sqrt{3}}{41472}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{33}{260} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{-19+114\,x}{192} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{-965+5790\,x}{1536}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}-{\frac{11}{16} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{429}{128}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}+{\frac{429\,\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 ) }-{\frac{1}{26} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}+{\frac{-1+6\,x}{52} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56935, size = 217, normalized size = 1.41 \begin{align*} \frac{1}{6} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x - \frac{7}{90} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{143}{144} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{737}{864} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}}}{4 \,{\left (2 \, x + 1\right )}} + \frac{5665}{1152} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{315623}{41472} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{429}{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{51997}{6912} \, \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 = 1.56708, size = 462, normalized size = 3. \begin{align*} \frac{1578115 \, \sqrt{3}{\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 694980 \, \sqrt{13}{\left (2 \, x + 1\right )} \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 ) + 12 \,{\left (103680 \, x^{6} - 65664 \, x^{5} + 251424 \, x^{4} - 115680 \, x^{3} + 310660 \, x^{2} - 322972 \, x - 364257\right )} \sqrt{3 \, x^{2} - x + 2}}{414720 \,{\left (2 \, 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 )}{\left (2 x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.01138, size = 1026, normalized size = 6.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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