Optimal. Leaf size=94 \[ \frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{484 \sqrt {23}}+\frac {2891 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{15004 \sqrt {31}}+\frac {3}{968} \log \left (3-x+2 x^2\right )-\frac {3}{968} \log \left (2+3 x+5 x^2\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {988, 1086,
648, 632, 210, 642} \begin {gather*} \frac {7 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{484 \sqrt {23}}+\frac {2891 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{15004 \sqrt {31}}+\frac {65 x+4}{682 \left (5 x^2+3 x+2\right )}+\frac {3}{968} \log \left (2 x^2-x+3\right )-\frac {3}{968} \log \left (5 x^2+3 x+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 988
Rule 1086
Rubi steps
\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1804+1397 x-1430 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{7502}\\ &=\frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {18755-22506 x}{3-x+2 x^2} \, dx}{1815484}-\frac {\int \frac {-158026+56265 x}{2+3 x+5 x^2} \, dx}{1815484}\\ &=\frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}+\frac {3}{968} \int \frac {-1+4 x}{3-x+2 x^2} \, dx-\frac {3}{968} \int \frac {3+10 x}{2+3 x+5 x^2} \, dx-\frac {7}{968} \int \frac {1}{3-x+2 x^2} \, dx+\frac {2891 \int \frac {1}{2+3 x+5 x^2} \, dx}{30008}\\ &=\frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}+\frac {3}{968} \log \left (3-x+2 x^2\right )-\frac {3}{968} \log \left (2+3 x+5 x^2\right )+\frac {7}{484} \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )-\frac {2891 \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{15004}\\ &=\frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{484 \sqrt {23}}+\frac {2891 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{15004 \sqrt {31}}+\frac {3}{968} \log \left (3-x+2 x^2\right )-\frac {3}{968} \log \left (2+3 x+5 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 94, normalized size = 1.00 \begin {gather*} \frac {4+65 x}{682 \left (2+3 x+5 x^2\right )}-\frac {7 \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )}{484 \sqrt {23}}+\frac {2891 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{15004 \sqrt {31}}+\frac {3}{968} \log \left (3-x+2 x^2\right )-\frac {3}{968} \log \left (2+3 x+5 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 77, normalized size = 0.82
method | result | size |
risch | \(\frac {\frac {13 x}{682}+\frac {2}{1705}}{x^{2}+\frac {3}{5} x +\frac {2}{5}}+\frac {3 \ln \left (16 x^{2}-8 x +24\right )}{968}-\frac {7 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{11132}-\frac {3 \ln \left (100 x^{2}+60 x +40\right )}{968}+\frac {2891 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{465124}\) | \(76\) |
default | \(\frac {3 \ln \left (2 x^{2}-x +3\right )}{968}-\frac {7 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{11132}-\frac {-\frac {286 x}{31}-\frac {88}{155}}{484 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {3 \ln \left (5 x^{2}+3 x +2\right )}{968}+\frac {2891 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{465124}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 78, normalized size = 0.83 \begin {gather*} \frac {2891}{465124} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {7}{11132} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {65 \, x + 4}{682 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {3}{968} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {3}{968} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.02, size = 117, normalized size = 1.24 \begin {gather*} \frac {132986 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - 13454 \, \sqrt {23} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 66309 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 66309 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (2 \, x^{2} - x + 3\right ) + 2039180 \, x + 125488}{21395704 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 102, normalized size = 1.09 \begin {gather*} \frac {65 x + 4}{3410 x^{2} + 2046 x + 1364} + \frac {3 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{968} - \frac {3 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{968} - \frac {7 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{11132} + \frac {2891 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{465124} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.38, size = 78, normalized size = 0.83 \begin {gather*} \frac {2891}{465124} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {7}{11132} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {65 \, x + 4}{682 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {3}{968} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {3}{968} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.57, size = 95, normalized size = 1.01 \begin {gather*} \frac {\frac {13\,x}{682}+\frac {2}{1705}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {3}{968}+\frac {\sqrt {23}\,7{}\mathrm {i}}{22264}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {3}{968}+\frac {\sqrt {23}\,7{}\mathrm {i}}{22264}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {3}{968}+\frac {\sqrt {31}\,2891{}\mathrm {i}}{930248}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {3}{968}+\frac {\sqrt {31}\,2891{}\mathrm {i}}{930248}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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