Optimal. Leaf size=129 \[ -\frac {325}{128} \log \left (x^2+5 x+1\right )+\frac {\sqrt {x^6+9 x^5+16 x^4-27 x^3-36 x^2-11 x-1} \left (16 x^3+104 x^2-6 x-185\right )}{64 \left (x^2+5 x+1\right )}+\frac {325}{128} \log \left (-2 x^3-9 x^2+2 \sqrt {x^6+9 x^5+16 x^4-27 x^3-36 x^2-11 x-1}+3 x+1\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.62, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6688, 6719, 1661, 640, 612, 621, 206} \begin {gather*} -\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 )}+\frac {325 \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )} \tanh ^{-1}\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx &=\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 ) \operatorname {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 )} \tanh ^{-1}\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*}
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Mathematica [A] time = 0.09, size = 101, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x^2-x-1} \left (x^2+5 x+1\right ) \left (325 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+2 \sqrt {x^2-x-1} \left (16 x^3+104 x^2-6 x-185\right )\right )}{128 \sqrt {\left (x^2-x-1\right ) \left (x^2+5 x+1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 129, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 139, normalized size = 1.08 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 97, normalized size = 0.75 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 107, normalized size = 0.83
method | result | size |
risch | \(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{64 x^{2}+320 x +64}-\frac {325 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\) | \(107\) |
default | \(\frac {\sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}\, \left (32 x \left (x^{2}-x -1\right )^{\frac {3}{2}}+240 \left (x^{2}-x -1\right )^{\frac {3}{2}}+260 x \sqrt {x^{2}-x -1}-130 \sqrt {x^{2}-x -1}-325 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )\right )}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\) | \(120\) |
trager | \(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}}{64 x^{2}+320 x +64}-\frac {325 \ln \left (\frac {2 x^{3}+9 x^{2}+2 \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}-3 x -1}{x^{2}+5 x +1}\right )}{128}\) | \(120\) |
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*} \int \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1}\,{d x} \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 \sqrt {x^6+9\,x^5+16\,x^4-27\,x^3-36\,x^2-11\,x-1} \,d x \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 \sqrt {x^{6} + 9 x^{5} + 16 x^{4} - 27 x^{3} - 36 x^{2} - 11 x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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