Optimal. Leaf size=70 \[ -\frac {5 \tan ^{-1}\left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {x^2+6 x-1}}\right )}{6 \sqrt {14}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {7} (x+1)}{\sqrt {x^2+6 x-1}}\right )}{3 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1035, 1029, 207, 203} \[ -\frac {5 \tan ^{-1}\left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {x^2+6 x-1}}\right )}{6 \sqrt {14}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {7} (x+1)}{\sqrt {x^2+6 x-1}}\right )}{3 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1029
Rule 1035
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx &=-\left (\frac {1}{42} \int \frac {-70-70 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx\right )+\frac {1}{42} \int \frac {-28+14 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx\\ &=-\left (\frac {896}{3} \operatorname {Subst}\left (\int \frac {1}{-200704+28 x^2} \, dx,x,\frac {-224-224 x}{\sqrt {-1+6 x+x^2}}\right )\right )-\frac {2800}{3} \operatorname {Subst}\left (\int \frac {1}{627200+28 x^2} \, dx,x,\frac {280-140 x}{\sqrt {-1+6 x+x^2}}\right )\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {-1+6 x+x^2}}\right )}{6 \sqrt {14}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {7} (1+x)}{\sqrt {-1+6 x+x^2}}\right )}{3 \sqrt {7}}\\ \end {align*}
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Mathematica [C] time = 0.45, size = 174, normalized size = 2.49 \[ -\frac {\sqrt {7-4 i \sqrt {2}} \left (8 \sqrt {2}+13 i\right ) \tan ^{-1}\left (\frac {\left (-7-2 i \sqrt {2}\right ) x-6 i \sqrt {2}+9}{\sqrt {7 \left (7-4 i \sqrt {2}\right )} \sqrt {x^2+6 x-1}}\right )+\sqrt {7+4 i \sqrt {2}} \left (8 \sqrt {2}-13 i\right ) \tan ^{-1}\left (\frac {\left (-7+2 i \sqrt {2}\right ) x+6 i \sqrt {2}+9}{\sqrt {7 \left (7+4 i \sqrt {2}\right )} \sqrt {x^2+6 x-1}}\right )}{108 \sqrt {14}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.30, size = 123, normalized size = 1.76 \[ \text {RootSum}\left [3 \text {$\#$1}^4-8 \text {$\#$1}^3+46 \text {$\#$1}^2-104 \text {$\#$1}+171\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^2+6 x-1}-x\right )-\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^2+6 x-1}-x\right )+4 \log \left (-\text {$\#$1}+\sqrt {x^2+6 x-1}-x\right )}{3 \text {$\#$1}^3-6 \text {$\#$1}^2+23 \text {$\#$1}-26}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 311, normalized size = 4.44 \[ \frac {1}{84} \, \sqrt {14} \sqrt {2} \log \left (13068 \, \sqrt {14} \sqrt {2} {\left (x - 2\right )} + 78408 \, x^{2} - 13068 \, \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} \sqrt {2} + 6 \, x + 4\right )} + 287496 \, x + 287496\right ) - \frac {1}{84} \, \sqrt {14} \sqrt {2} \log \left (-13068 \, \sqrt {14} \sqrt {2} {\left (x - 2\right )} + 78408 \, x^{2} + 13068 \, \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} \sqrt {2} - 6 \, x - 4\right )} + 287496 \, x + 287496\right ) - \frac {5}{42} \, \sqrt {14} \arctan \left (\frac {1}{24} \, \sqrt {3} \sqrt {\sqrt {14} \sqrt {2} {\left (x - 2\right )} + 6 \, x^{2} - \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} \sqrt {2} + 6 \, x + 4\right )} + 22 \, x + 22} {\left (\sqrt {14} + \sqrt {2}\right )} + \frac {1}{8} \, \sqrt {2} {\left (x + 3\right )} + \frac {1}{8} \, \sqrt {14} {\left (x + 1\right )} - \frac {1}{8} \, \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} + \sqrt {2}\right )}\right ) - \frac {5}{42} \, \sqrt {14} \arctan \left (-\frac {1}{8} \, \sqrt {2} {\left (x + 3\right )} + \frac {1}{8} \, \sqrt {14} {\left (x + 1\right )} + \frac {1}{1584} \, \sqrt {-13068 \, \sqrt {14} \sqrt {2} {\left (x - 2\right )} + 78408 \, x^{2} + 13068 \, \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} \sqrt {2} - 6 \, x - 4\right )} + 287496 \, x + 287496} {\left (\sqrt {14} - \sqrt {2}\right )} - \frac {1}{8} \, \sqrt {x^{2} + 6 \, x - 1} {\left (\sqrt {14} - \sqrt {2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 257, normalized size = 3.67 \[ -\frac {5}{84} \, \sqrt {7} \sqrt {2} {\left (\arctan \relax (2) + \arctan \left (\frac {1}{8} \, {\left (x - \sqrt {x^{2} + 6 \, x - 1}\right )} {\left (\sqrt {14} + \sqrt {2}\right )} + \frac {1}{8} \, \sqrt {14} + \frac {3}{8} \, \sqrt {2}\right )\right )} + \frac {5}{84} \, \sqrt {7} \sqrt {2} {\left (\arctan \left (\frac {1}{2}\right ) + \arctan \left (-\frac {1}{8} \, {\left (x - \sqrt {x^{2} + 6 \, x - 1}\right )} {\left (\sqrt {14} - \sqrt {2}\right )} - \frac {1}{8} \, \sqrt {14} + \frac {3}{8} \, \sqrt {2}\right )\right )} + \frac {1}{42} \, \sqrt {7} \log \left (4 \, {\left (4 \, \sqrt {7} \sqrt {2} + 3 \, x + \sqrt {7} - 4 \, \sqrt {2} - 3 \, \sqrt {x^{2} + 6 \, x - 1} + 2\right )}^{2} + 16 \, {\left (\sqrt {7} \sqrt {2} - 3 \, x - \sqrt {7} - \sqrt {2} + 3 \, \sqrt {x^{2} + 6 \, x - 1} - 2\right )}^{2}\right ) - \frac {1}{42} \, \sqrt {7} \log \left (4 \, {\left (4 \, \sqrt {7} \sqrt {2} + 3 \, x - \sqrt {7} + 4 \, \sqrt {2} - 3 \, \sqrt {x^{2} + 6 \, x - 1} + 2\right )}^{2} + 16 \, {\left (\sqrt {7} \sqrt {2} - 3 \, x + \sqrt {7} + \sqrt {2} + 3 \, \sqrt {x^{2} + 6 \, x - 1} - 2\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.95, size = 158, normalized size = 2.26
method | result | size |
default | \(-\frac {\sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \left (4 \sqrt {7}\, \arctanh \left (\frac {\sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \sqrt {7}}{21}\right )-5 \sqrt {14}\, \arctan \left (\frac {\sqrt {14}\, \sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \left (-2+x \right )}{4 \left (\frac {2 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}-5\right ) \left (-1-x \right )}\right )\right )}{84 \sqrt {-\frac {3 \left (\frac {2 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}-5\right )}{\left (1+\frac {-2+x}{-1-x}\right )^{2}}}\, \left (1+\frac {-2+x}{-1-x}\right )}\) | \(158\) |
trager | \(\RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) \ln \left (-\frac {1568802816 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5}+6019776 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x +39686976 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+3171168 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}+5768 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +73542 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+50611 \sqrt {x^{2}+6 x -1}}{2016 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}-3 x -40}\right )-\frac {672 \ln \left (\frac {159860736 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5}+2221632 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x -4044096 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+352352 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}-2340 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +3978 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+319 \sqrt {x^{2}+6 x -1}}{2016 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}+37 x +40}\right ) \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}}{11}-\frac {34 \ln \left (\frac {159860736 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5}+2221632 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x -4044096 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+352352 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}-2340 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +3978 \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+319 \sqrt {x^{2}+6 x -1}}{2016 x \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}+37 x +40}\right ) \RootOf \left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )}{33}\) | \(502\) |
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 \[ \int \frac {2 \, x + 1}{{\left (3 \, x^{2} + 4 \, x + 4\right )} \sqrt {x^{2} + 6 \, x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {2\,x+1}{\sqrt {x^2+6\,x-1}\,\left (3\,x^2+4\,x+4\right )} \,d x \]
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 \[ \int \frac {2 x + 1}{\sqrt {x^{2} + 6 x - 1} \left (3 x^{2} + 4 x + 4\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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