Integrand size = 32, antiderivative size = 129 \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\frac {\left (-185-6 x+104 x^2+16 x^3\right ) \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}}{64 \left (1+5 x+x^2\right )}-\frac {325}{128} \log \left (1+5 x+x^2\right )+\frac {325}{128} \log \left (1+3 x-9 x^2-2 x^3+2 \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.62, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6820, 6851, 1675, 654, 626, 635, 212} \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\frac {325 \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{128 \sqrt {x^2-x-1} \left (x^2+5 x+1\right )}-\frac {65 \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )} (1-2 x)}{64 \left (x^2+5 x+1\right )}-\frac {x \left (-x^2+x+1\right ) \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )}}{4 \left (x^2+5 x+1\right )}-\frac {15 \left (-x^2+x+1\right ) \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )}}{8 \left (x^2+5 x+1\right )} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1675
Rule 6820
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \, dx \\ & = \frac {\sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \int \sqrt {-1-x+x^2} \left (1+5 x+x^2\right ) \, dx}{\sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ & = -\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {\sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \int \left (5+\frac {45 x}{2}\right ) \sqrt {-1-x+x^2} \, dx}{4 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ & = -\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {\left (65 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \int \sqrt {-1-x+x^2} \, dx}{16 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ & = -\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}-\frac {\left (325 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{128 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ & = -\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}-\frac {\left (325 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{64 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ & = -\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {325 \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{128 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77 \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\frac {\sqrt {-1-x+x^2} \left (1+5 x+x^2\right ) \left (2 \sqrt {-1-x+x^2} \left (-185-6 x+104 x^2+16 x^3\right )+325 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )\right )}{128 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}} \]
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Time = 1.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{64 x^{2}+320 x +64}-\frac {325 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\) | \(107\) |
default | \(-\frac {\sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}\, \left (-32 x \left (x^{2}-x -1\right )^{\frac {3}{2}}-240 \left (x^{2}-x -1\right )^{\frac {3}{2}}-260 x \sqrt {x^{2}-x -1}+325 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )+130 \sqrt {x^{2}-x -1}\right )}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\) | \(120\) |
trager | \(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}}{64 x^{2}+320 x +64}+\frac {325 \ln \left (\frac {1+3 x -9 x^{2}-2 x^{3}+2 \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}}{x^{2}+5 x +1}\right )}{128}\) | \(120\) |
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Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08 \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\frac {569 \, x^{2} + 2600 \, {\left (x^{2} + 5 \, x + 1\right )} \log \left (-\frac {2 \, x^{3} + 9 \, x^{2} - 3 \, x - 2 \, \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1} - 1}{x^{2} + 5 \, x + 1}\right ) + 16 \, \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1} {\left (16 \, x^{3} + 104 \, x^{2} - 6 \, x - 185\right )} + 2845 \, x + 569}{1024 \, {\left (x^{2} + 5 \, x + 1\right )}} \]
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\[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\int \sqrt {x^{6} + 9 x^{5} + 16 x^{4} - 27 x^{3} - 36 x^{2} - 11 x - 1}\, dx \]
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\[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\int { \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.75 \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\frac {325}{128} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right ) + \frac {1}{64} \, {\left (2 \, {\left (4 \, {\left (2 \, x \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right ) + 13 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} x - 3 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} x - 185 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} \sqrt {x^{2} - x - 1} \]
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Timed out. \[ \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx=\int \sqrt {x^6+9\,x^5+16\,x^4-27\,x^3-36\,x^2-11\,x-1} \,d x \]
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