Optimal. Leaf size=154 \[ -\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac {315623 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{13824 \sqrt {3}}+\frac {429}{128} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right ) \]
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Rubi [A]
time = 0.10, 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 = {1664, 828, 857,
633, 221, 738, 212} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 828
Rule 857
Rule 1664
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} \text {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )+\frac {315623 \text {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 = 0.64, size = 131, normalized size = 0.85 \begin {gather*} \frac {\frac {6 \sqrt {2-x+3 x^2} \left (-364257-322972 x+310660 x^2-115680 x^3+251424 x^4-65664 x^5+103680 x^6\right )}{1+2 x}-1389960 \sqrt {13} \tanh ^{-1}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )-1578115 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{207360} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 235, normalized size = 1.53
method | result | size |
risch | \(\frac {311040 x^{8}-300672 x^{7}+1027296 x^{6}-729792 x^{5}+1550508 x^{4}-1510936 x^{3}-148479 x^{2}-281687 x -728514}{34560 \left (2 x +1\right ) \sqrt {3 x^{2}-x +2}}+\frac {315623 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{41472}+\frac {429 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{128}\) | \(107\) |
trager | \(\frac {\left (103680 x^{6}-65664 x^{5}+251424 x^{4}-115680 x^{3}+310660 x^{2}-322972 x -364257\right ) \sqrt {3 x^{2}-x +2}}{69120 x +34560}-\frac {315623 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}+\RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{41472}-\frac {429 \RootOf \left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \RootOf \left (\textit {\_Z}^{2}-13\right )}{2 x +1}\right )}{128}\) | \(141\) |
default | \(\frac {\left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{36}+\frac {115 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{1728}+\frac {2645 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{13824}+\frac {315623 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{41472}-\frac {33 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{260}+\frac {19 \left (6 x -1\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{192}+\frac {965 \left (6 x -1\right ) \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}{1536}-\frac {11 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{16}-\frac {429 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}{128}+\frac {429 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{128}-\frac {\left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {7}{2}}}{26 \left (x +\frac {1}{2}\right )}+\frac {\left (6 x -1\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{52}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 161, normalized size = 1.05 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 153, normalized size = 0.99 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\left (3 x^{2} - x + 2\right )^{\frac {5}{2}} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 760 vs.
\(2 (123) = 246\).
time = 6.41, size = 760, normalized size = 4.94 \begin {gather*} \frac {429}{128} \, \sqrt {13} \log \left (\sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} - 4\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {315623}{41472} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {2 \, \sqrt {13}}{2 \, x + 1} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {169}{128} \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + \frac {5154065 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{11} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 7837020 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{10} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 39468815 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 14445540 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 460893402 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 343084680 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 944150094 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 22871160 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 1397032245 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 683367516 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 392684355 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 197538588 \, \sqrt {13} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}{34560 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{2} - 3\right )}^{6}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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