Optimal. Leaf size=174 \[ \frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1030, 1046,
738, 212} \begin {gather*} \frac {2 (22 x+21)}{5 \sqrt {2 x^2+3 x+1}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )+\frac {1}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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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 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {2}{15} \int \frac {-72+\frac {81 x}{2}}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx\\ &=\frac {2 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{5} \left (9 \left (3-\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 (3+\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 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}+\frac {1}{5} \left (18 \left (3-\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 (3+\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 (21+22 x)}{5 \sqrt {1+3 x+2 x^2}}-\frac {1}{10} \sqrt {\frac {3}{5} \left (2065+653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{10} \sqrt {\frac {3}{5} \left (2065-653 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 130, normalized size = 0.75 \begin {gather*} \frac {1}{25} \left (\frac {5 (42+44 x)}{\sqrt {1+3 x+2 x^2}}-\sqrt {30975+9795 \sqrt {10}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )}{\sqrt {2065+653 \sqrt {10}}}\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. \(465\) vs.
\(2(122)=244\).
time = 0.62, size = 466, normalized size = 2.68 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. 668 vs.
\(2 (122) = 244\).
time = 0.53, size = 668, normalized size = 3.84 \begin {gather*} -\frac {1}{60} \, \sqrt {10} {\left (\frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {588 \, \sqrt {10} x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {2112 \, x}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {27 \, \sqrt {10} \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} - \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {450 \, \sqrt {10}}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} - \frac {216 \, \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (17 \, \sqrt {10} + 55\right )}^{\frac {3}{2}}} + \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{{\left (-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}\right )}^{\frac {3}{2}}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} + 55 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + \frac {1656}{17 \, \sqrt {10} \sqrt {2 \, x^{2} + 3 \, x + 1} - 55 \, \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. 365 vs.
\(2 (122) = 244\).
time = 0.36, size = 365, normalized size = 2.10 \begin {gather*} \frac {\sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {1959 \, \sqrt {10} + 6195} \log \left (-\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x - 130 \, \sqrt {5} x\right )} \sqrt {1959 \, \sqrt {10} + 6195} + 90 \, x - 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 90}{x}\right ) + \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x + {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) - \sqrt {5} {\left (2 \, x^{2} + 3 \, x + 1\right )} \sqrt {-1959 \, \sqrt {10} + 6195} \log \left (\frac {45 \, \sqrt {10} x - {\left (41 \, \sqrt {10} \sqrt {5} x + 130 \, \sqrt {5} x\right )} \sqrt {-1959 \, \sqrt {10} + 6195} - 90 \, x + 90 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 90}{x}\right ) + 840 \, x^{2} + 20 \, \sqrt {2 \, x^{2} + 3 \, x + 1} {\left (22 \, x + 21\right )} + 1260 \, x + 420}{50 \, {\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} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 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} + x^{3} \sqrt {2 x^{2} + 3 x + 1} - 13 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 [A]
time = 4.42, size = 112, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (22 \, x + 21\right )}}{5 \, \sqrt {2 \, x^{2} + 3 \, x + 1}} + 0.0140045514133333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 4.97793168620000 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.0140045514125333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \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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