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 = 0.16, antiderivative size = 154, normalized size of antiderivative = 1.00, 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 206
Rule 215
Rule 619
Rule 724
Rule 814
Rule 843
Rule 1650
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*}
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Mathematica [A] time = 0.12, size = 113, normalized size = 0.73 \[ \frac {694980 \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )+\frac {6 \sqrt {3 x^2-x+2} \left (103680 x^6-65664 x^5+251424 x^4-115680 x^3+310660 x^2-322972 x-364257\right )}{2 x+1}+1578115 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{207360} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 153, normalized size = 0.99 \[ \frac {1578115 \, \sqrt {3} {\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 694980 \, \sqrt {13} {\left (2 \, x + 1\right )} \log \left (\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 12 \, {\left (103680 \, x^{6} - 65664 \, x^{5} + 251424 \, x^{4} - 115680 \, x^{3} + 310660 \, x^{2} - 322972 \, x - 364257\right )} \sqrt {3 \, x^{2} - x + 2}}{414720 \, {\left (2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 760, normalized size = 4.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 235, normalized size = 1.53 \[ \frac {315623 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{41472}+\frac {429 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{128}+\frac {\left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{36}+\frac {115 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{1728}+\frac {2645 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{13824}-\frac {33 \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {5}{2}}}{260}+\frac {19 \left (6 x -1\right ) \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}{192}+\frac {965 \left (6 x -1\right ) \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}{1536}-\frac {11 \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}{16}-\frac {429 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}{128}-\frac {\left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {7}{2}}}{26 \left (x +\frac {1}{2}\right )}+\frac {\left (6 x -1\right ) \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {5}{2}}}{52} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 161, normalized size = 1.05 \[ \frac {1}{6} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x - \frac {7}{90} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {143}{144} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {737}{864} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {{\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 1\right )}} + \frac {5665}{1152} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {315623}{41472} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {429}{128} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) - \frac {51997}{6912} \, \sqrt {3 \, x^{2} - x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x^2-x+2\right )}^{5/2}\,\left (4\,x^2+3\,x+1\right )}{{\left (2\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x^{2} - x + 2\right )^{\frac {5}{2}} \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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