Optimal. Leaf size=166 \[ -\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}-\frac {9}{2} \sqrt {\frac {1}{5} \left (-3+\sqrt {10}\right )} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 166, 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, 12, 1046,
738, 210, 212} \begin {gather*} -\frac {9}{2} \sqrt {\frac {1}{5} \left (\sqrt {10}-3\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)}{17 \sqrt {-2 x^2+3 x+1}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (3+\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 12
Rule 210
Rule 212
Rule 738
Rule 1030
Rule 1046
Rubi steps
\begin {align*} \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{3/2}} \, dx &=-\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+\frac {2}{17} \int \frac {153 x}{2 \left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+9 \int \frac {x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx\\ &=-\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+\frac {1}{5} \left (9 \left (5-\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 (9 \left (5+\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)}{17 \sqrt {1+3 x-2 x^2}}-\frac {1}{5} \left (18 \left (5-\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 (18 \left (5+\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)}{17 \sqrt {1+3 x-2 x^2}}-\frac {9}{2} \sqrt {\frac {1}{5} \left (-3+\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 (3+\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 = 0.32, size = 137, normalized size = 0.83 \begin {gather*} -\frac {2 (15+14 x)}{17 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \text {RootSum}\left [5+20 \text {$\#$1}+8 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {3 \log (x)-3 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right )-2 \log (x) \text {$\#$1}+2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}}{5+4 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$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. \(454\) vs.
\(2(118)=236\).
time = 0.57, size = 455, normalized size = 2.74 Too large to display
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. 678 vs.
\(2 (118) = 236\).
time = 0.52, size = 678, normalized size = 4.08 \begin {gather*} \frac {1}{340} \, \sqrt {10} {\left (\frac {124 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {124 \, \sqrt {10} x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {153 \, \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 )}{\sqrt {10} \sqrt {\sqrt {10} + 1} + \sqrt {\sqrt {10} + 1}} - \frac {128 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {128 \, x}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {1224 \, \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 )}{\sqrt {10} \sqrt {\sqrt {10} + 1} + \sqrt {\sqrt {10} + 1}} + \frac {153 \, \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 {3}{2}}} - \frac {42 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {42 \, \sqrt {10}}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}} + \frac {1224 \, \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 {3}{2}}} - \frac {312}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} + \sqrt {-2 \, x^{2} + 3 \, x + 1}} - \frac {312}{\sqrt {10} \sqrt {-2 \, x^{2} + 3 \, x + 1} - \sqrt {-2 \, x^{2} + 3 \, x + 1}}\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. 344 vs.
\(2 (118) = 236\).
time = 0.40, size = 344, normalized size = 2.07 \begin {gather*} -\frac {612 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} - 3} \arctan \left (\frac {\sqrt {10} \sqrt {5} \sqrt {2} x \sqrt {\sqrt {10} - 3} \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 (x + 1\right )} - \sqrt {10} \sqrt {5} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 5 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} - 3}}{10 \, x}\right ) + 153 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} + 3} \log \left (\frac {9 \, {\left (5 \, \sqrt {10} x + {\left (3 \, \sqrt {10} \sqrt {5} x - 10 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} + 3} - 10 \, x + 10 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 10\right )}}{x}\right ) - 153 \, \sqrt {5} {\left (2 \, x^{2} - 3 \, x - 1\right )} \sqrt {\sqrt {10} + 3} \log \left (\frac {9 \, {\left (5 \, \sqrt {10} x - {\left (3 \, \sqrt {10} \sqrt {5} x - 10 \, \sqrt {5} x\right )} \sqrt {\sqrt {10} + 3} - 10 \, x + 10 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 10\right )}}{x}\right ) + 600 \, x^{2} - 20 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} {\left (14 \, x + 15\right )} - 900 \, x - 300}{170 \, {\left (2 \, x^{2} - 3 \, 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}{- 6 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 17 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 5 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 10 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{- 6 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 17 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 5 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 10 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 )}^{3/2}\,\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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