Optimal. Leaf size=63 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {5+4 x+4 x^2}}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {11}{15}} (1+2 x)}{\sqrt {5+4 x+4 x^2}}\right )}{\sqrt {165}} \]
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Rubi [A]
time = 0.04, 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 = {1039, 996, 213,
1038, 210} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 213
Rule 996
Rule 1038
Rule 1039
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 \text {Subst}\left (\int \frac {1}{-240+11 x^2} \, dx,x,\frac {4+8 x}{\sqrt {5+4 x+4 x^2}}\right )-\text {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] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.12, size = 102, normalized size = 1.62 \begin {gather*} \frac {1}{2} \text {RootSum}\left [69-108 \text {$\#$1}+58 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-5 \log \left (-2 x+\sqrt {5+4 x+4 x^2}-\text {$\#$1}\right )+\log \left (-2 x+\sqrt {5+4 x+4 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-27+29 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.49, 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 | \(-\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 +9405 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+97185 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}+3784 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x +256 \sqrt {4 x^{2}+4 x +5}+1364 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )}{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 +9405 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+97185 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}+3784 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x +256 \sqrt {4 x^{2}+4 x +5}+1364 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )}{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}-\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 +3420 \sqrt {4 x^{2}+4 x +5}\, \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+41385 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{3}-754 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right ) x +52 \sqrt {4 x^{2}+4 x +5}+899 \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )}{165 x \RootOf \left (27225 \textit {\_Z}^{4}+1155 \textit {\_Z}^{2}+16\right )^{2}+3 x -4}\right )\) | \(515\) |
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 \begin {gather*} \text {Failed to integrate} \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. 307 vs.
\(2 (52) = 104\).
time = 0.34, size = 307, normalized size = 4.87 \begin {gather*} \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 ) \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}{\left (x^{2} + x + 4\right ) \sqrt {4 x^{2} + 4 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs.
\(2 (52) = 104\).
time = 0.02, size = 292, normalized size = 4.63 \begin {gather*} 2 \left (\frac {1}{660} \sqrt {165} \ln \left (\left (300 \left (\sqrt {4 x^{2}+4 x+5}-2 x\right )-300\right ) \left (300 \left (\sqrt {4 x^{2}+4 x+5}-2 x\right )-300\right )+\left (300 \sqrt {11}-300 \sqrt {15}\right ) \left (300 \sqrt {11}-300 \sqrt {15}\right )\right )-\frac {1}{330} \sqrt {165} \sqrt {15} \arctan \left (\frac {-4 \left (\sqrt {4 x^{2}+4 x+5}-2 x\right )+4}{4 \sqrt {11}-4 \sqrt {15}}\right )-\frac {1}{660} \sqrt {165} \ln \left (\left (300 \left (\sqrt {4 x^{2}+4 x+5}-2 x\right )-300\right ) \left (300 \left (\sqrt {4 x^{2}+4 x+5}-2 x\right )-300\right )+\left (-300 \sqrt {11}-300 \sqrt {15}\right ) \left (-300 \sqrt {11}-300 \sqrt {15}\right )\right )+\frac {1}{330} \sqrt {165} \sqrt {15} \arctan \left (\frac {-\sqrt {4 x^{2}+4 x+5}+2 x+1}{-\sqrt {11}-\sqrt {15}}\right )\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 = -0.02 \begin {gather*} \int \frac {x}{\sqrt {4\,x^2+4\,x+5}\,\left (x^2+x+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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