Optimal. Leaf size=193 \[ -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1030, 1074,
1046, 738, 210, 212} \begin {gather*} \frac {9}{2} \sqrt {\frac {1}{5} \left (17 \sqrt {10}-53\right )} \text {ArcTan}\left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac {2 (4814 x+291)}{867 \sqrt {-2 x^2+3 x+1}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 738
Rule 1030
Rule 1046
Rule 1074
Rubi steps
\begin {align*} \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}+\frac {2}{51} \int \frac {-56+\frac {235 x}{2}+84 x^2}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {4}{867} \int \frac {\frac {7803}{2}+\frac {23409 x}{4}}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {1}{5} \left (27 \left (5-2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx+\frac {1}{5} \left (27 \left (5+2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}-\frac {1}{5} \left (54 \left (5-2 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4-2 \sqrt {10}\right )-8 \left (4-2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4-2 \sqrt {10}\right )-\left (18-4 \left (4-2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )-\frac {1}{5} \left (54 \left (5+2 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4+2 \sqrt {10}\right )-8 \left (4+2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4+2 \sqrt {10}\right )-\left (18-4 \left (4+2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )\\ &=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.46, size = 183, normalized size = 0.95 \begin {gather*} -\frac {2 \left (546+5925 x+13860 x^2-9628 x^3\right )}{867 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {9}{2} \text {RootSum}\left [5+20 \text {$\#$1}+8 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-13 \log (x)+13 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}-6 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^2+2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{5+4 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(867\) vs.
\(2(137)=274\).
time = 0.61, size = 868, normalized size = 4.50
method | result | size |
trager | \(\frac {2 \left (9628 x^{3}-13860 x^{2}-5925 x -546\right ) \sqrt {-2 x^{2}+3 x +1}}{867 \left (2 x^{2}-3 x -1\right )^{2}}-18 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) \ln \left (-\frac {-2105600 x \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{5}+5362400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3} x +74880 \sqrt {-2 x^{2}+3 x +1}\, \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}+473760 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3}-3406349 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) x -99945 \sqrt {-2 x^{2}+3 x +1}-632106 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )}{80 x \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-87 x -34}\right )-\frac {9 \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) \ln \left (\frac {2105600 \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{4} x -217440 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +1497600 \sqrt {-2 x^{2}+3 x +1}\, \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}+473760 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-2187 \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +14580 \sqrt {-2 x^{2}+3 x +1}+4374 \RootOf \left (\textit {\_Z}^{2}+400 \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )}{80 x \RootOf \left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-19 x +34}\right )}{10}\) | \(483\) |
default | \(\text {Expression too large to display}\) | \(868\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1276 vs.
\(2 (137) = 274\).
time = 0.56, size = 1276, normalized size = 6.61 \begin {gather*} \frac {1}{17340} \, \sqrt {10} {\left (\frac {2108 \, \sqrt {10} x}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} + {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} - \frac {2108 \, \sqrt {10} x}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} - {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} - \frac {56916 \, \sqrt {10} x}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {56916 \, \sqrt {10} x}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1984 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {1984 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {70227 \, \sqrt {10} \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{2 \, \sqrt {10} \sqrt {\sqrt {10} + 1} + 11 \, \sqrt {\sqrt {10} + 1}} - \frac {2176 \, x}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} + {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} - \frac {2176 \, x}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} - {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + \frac {58752 \, x}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {58752 \, x}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {2048 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {2048 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {561816 \, \arcsin \left (\frac {8 \, \sqrt {17} \sqrt {10} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {17} x}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {6 \, \sqrt {17} \sqrt {10}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {24 \, \sqrt {17}}{17 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right )}{2 \, \sqrt {10} \sqrt {\sqrt {10} + 1} + 11 \, \sqrt {\sqrt {10} + 1}} - \frac {714 \, \sqrt {10}}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} + {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + \frac {714 \, \sqrt {10}}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} - {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + \frac {19278 \, \sqrt {10}}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {19278 \, \sqrt {10}}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {1488 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1488 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {5304}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} + {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} - \frac {5304}{\sqrt {10} {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}} - {\left (-2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + \frac {143208}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {143208}{2 \, \sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - 11 \, \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1536}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1536}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {70227 \, \sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{{\left (\sqrt {10} - 1\right )}^{\frac {5}{2}}} + \frac {561816 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} \sqrt {\sqrt {10} - 1}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {2 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} - \frac {2}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {1}{18}\right )}{{\left (\sqrt {10} - 1\right )}^{\frac {5}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs.
\(2 (137) = 274\).
time = 0.43, size = 439, normalized size = 2.27 \begin {gather*} -\frac {43680 \, x^{4} - 131040 \, x^{3} - 31212 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} - 53} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {10} \sqrt {5} x + 10 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} - 53} \sqrt {\frac {6 \, x^{2} + \sqrt {10} {\left (3 \, x^{2} + 2 \, x\right )} - 2 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (\sqrt {10} x + 2 \, x + 2\right )} + 10 \, x + 4}{x^{2}}} + 2 \, {\left (\sqrt {10} \sqrt {5} {\left (6 \, x + 1\right )} - \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (\sqrt {10} \sqrt {5} + 10 \, \sqrt {5}\right )} + 5 \, \sqrt {5} {\left (3 \, x + 2\right )}\right )} \sqrt {17 \, \sqrt {10} - 53}}{90 \, x}\right ) - 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 54600 \, x^{2} - 20 \, {\left (9628 \, x^{3} - 13860 \, x^{2} - 5925 \, x - 546\right )} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 65520 \, x + 10920}{8670 \, {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, 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 {x}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {x+2}{{\left (-2\,x^2+3\,x+1\right )}^{5/2}\,\left (-3\,x^2+4\,x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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