Optimal. Leaf size=63 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {4 x^2+4 x+5}}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {11}{15}} (2 x+1)}{\sqrt {4 x^2+4 x+5}}\right )}{\sqrt {165}} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1025, 982, 207, 1024, 204} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {4 x^2+4 x+5}}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {11}{15}} (2 x+1)}{\sqrt {4 x^2+4 x+5}}\right )}{\sqrt {165}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 207
Rule 982
Rule 1024
Rule 1025
Rubi steps
\begin {align*} \int \frac {x}{\left (4+x+x^2\right ) \sqrt {5+4 x+4 x^2}} \, dx &=\frac {1}{8} \int \frac {4+8 x}{\left (4+x+x^2\right ) \sqrt {5+4 x+4 x^2}} \, dx-\frac {1}{2} \int \frac {1}{\left (4+x+x^2\right ) \sqrt {5+4 x+4 x^2}} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {1}{-240+11 x^2} \, dx,x,\frac {4+8 x}{\sqrt {5+4 x+4 x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,\sqrt {5+4 x+4 x^2}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {5+4 x+4 x^2}}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {11}{15}} (1+2 x)}{\sqrt {5+4 x+4 x^2}}\right )}{\sqrt {165}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 114, normalized size = 1.81 \[ \frac {\left (\sqrt {15}-i\right ) \tan ^{-1}\left (\frac {-2 i \sqrt {15} x-i \sqrt {15}+4}{\sqrt {11} \sqrt {4 x^2+4 x+5}}\right )+\left (\sqrt {15}+i\right ) \tan ^{-1}\left (\frac {2 i \sqrt {15} x+i \sqrt {15}+4}{\sqrt {11} \sqrt {4 x^2+4 x+5}}\right )}{2 \sqrt {165}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.24, size = 102, normalized size = 1.62 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3+58 \text {$\#$1}^2-108 \text {$\#$1}+69\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {4 x^2+4 x+5}-2 x\right )-5 \log \left (-\text {$\#$1}+\sqrt {4 x^2+4 x+5}-2 x\right )}{\text {$\#$1}^3-3 \text {$\#$1}^2+29 \text {$\#$1}-27}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 307, normalized size = 4.87 \[ \frac {2}{165} \, \sqrt {165} \sqrt {15} \arctan \left (\frac {1}{60} \, \sqrt {2} \sqrt {4 \, x^{2} - \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (2 \, x + 1\right )} + 4 \, x - \sqrt {165} + 16} {\left (\sqrt {165} \sqrt {15} + 15 \, \sqrt {15}\right )} + \frac {1}{60} \, \sqrt {165} \sqrt {15} {\left (2 \, x + 1\right )} - \frac {1}{60} \, \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (\sqrt {165} \sqrt {15} + 15 \, \sqrt {15}\right )} + \frac {1}{4} \, \sqrt {15} {\left (2 \, x + 1\right )}\right ) + \frac {2}{165} \, \sqrt {165} \sqrt {15} \arctan \left (\frac {1}{60} \, \sqrt {2} \sqrt {4 \, x^{2} - \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (2 \, x + 1\right )} + 4 \, x + \sqrt {165} + 16} {\left (\sqrt {165} \sqrt {15} - 15 \, \sqrt {15}\right )} + \frac {1}{60} \, \sqrt {165} \sqrt {15} {\left (2 \, x + 1\right )} - \frac {1}{60} \, \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (\sqrt {165} \sqrt {15} - 15 \, \sqrt {15}\right )} - \frac {1}{4} \, \sqrt {15} {\left (2 \, x + 1\right )}\right ) - \frac {1}{330} \, \sqrt {165} \log \left (460800 \, x^{2} - 115200 \, \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (2 \, x + 1\right )} + 460800 \, x + 115200 \, \sqrt {165} + 1843200\right ) + \frac {1}{330} \, \sqrt {165} \log \left (460800 \, x^{2} - 115200 \, \sqrt {4 \, x^{2} + 4 \, x + 5} {\left (2 \, x + 1\right )} + 460800 \, x - 115200 \, \sqrt {165} + 1843200\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 165, normalized size = 2.62 \[ \frac {1}{165} \, \sqrt {165} \sqrt {15} \arctan \left (-\frac {2 \, x - \sqrt {4 \, x^{2} + 4 \, x + 5} + 1}{\sqrt {15} + \sqrt {11}}\right ) - \frac {1}{165} \, \sqrt {165} \sqrt {15} \arctan \left (-\frac {2 \, x - \sqrt {4 \, x^{2} + 4 \, x + 5} + 1}{\sqrt {15} - \sqrt {11}}\right ) - \frac {1}{330} \, \sqrt {165} \log \left (90000 \, {\left (2 \, x - \sqrt {4 \, x^{2} + 4 \, x + 5} + 1\right )}^{2} + 90000 \, {\left (\sqrt {15} + \sqrt {11}\right )}^{2}\right ) + \frac {1}{330} \, \sqrt {165} \log \left (90000 \, {\left (2 \, x - \sqrt {4 \, x^{2} + 4 \, x + 5} + 1\right )}^{2} + 90000 \, {\left (\sqrt {15} - \sqrt {11}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 53, normalized size = 0.84
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {4 x^{2}+4 x +5}\, \sqrt {11}}{11}\right ) \sqrt {11}}{11}-\frac {\sqrt {165}\, \arctanh \left (\frac {\sqrt {165}\, \left (8 x +4\right )}{60 \sqrt {4 x^{2}+4 x +5}}\right )}{165}\) | \(53\) |
trager | \(\RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) \ln \left (-\frac {3524400 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{5} x +111270 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3} x -41385 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}+3420 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+754 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x -899 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )+52 \sqrt {4 x^{2}+4 x +5}}{165 x \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+3 x -4}\right )+\frac {165 \ln \left (-\frac {8276400 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{5} x +385770 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3} x +97185 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}-9405 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+3784 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x +1364 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )-256 \sqrt {4 x^{2}+4 x +5}}{165 x \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+4 x +4}\right ) \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}}{4}+\frac {7 \ln \left (-\frac {8276400 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{5} x +385770 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3} x +97185 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}-9405 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+3784 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x +1364 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )-256 \sqrt {4 x^{2}+4 x +5}}{165 x \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+4 x +4}\right ) \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )}{4}\) | \(516\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {4 \, x^{2} + 4 \, x + 5} {\left (x^{2} + x + 4\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{\sqrt {4\,x^2+4\,x+5}\,\left (x^2+x+4\right )} \,d x \]
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 \[ \int \frac {x}{\left (x^{2} + x + 4\right ) \sqrt {4 x^{2} + 4 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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