Integrand size = 20, antiderivative size = 71 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-x+2 \sqrt {3} \sqrt {-3+2 x}-21 \sqrt {\frac {3}{2}} \arctan \left (\frac {3+\sqrt {-9+6 x}}{2 \sqrt {6}}\right )+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1642, 648, 632, 210, 642} \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-21 \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right )-x+2 \sqrt {3} \sqrt {2 x-3}+10 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right ) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1642
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {x \left (-63+x^2\right )}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (-6+x+\frac {6 (33-10 x)}{33+6 x+x^2}\right ) \, dx,x,\sqrt {-9+6 x}\right )\right ) \\ & = -x+2 \sqrt {3} \sqrt {-3+2 x}-2 \text {Subst}\left (\int \frac {33-10 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right ) \\ & = -x+2 \sqrt {3} \sqrt {-3+2 x}+10 \text {Subst}\left (\int \frac {6+2 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )-126 \text {Subst}\left (\int \frac {1}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right ) \\ & = -x+2 \sqrt {3} \sqrt {-3+2 x}+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )+252 \text {Subst}\left (\int \frac {1}{-96-x^2} \, dx,x,6+2 \sqrt {-9+6 x}\right ) \\ & = -x+2 \sqrt {3} \sqrt {-3+2 x}-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {3+\sqrt {3} \sqrt {-3+2 x}}{2 \sqrt {6}}\right )+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=\frac {1}{2} \left (3-2 x+4 \sqrt {-9+6 x}-21 \sqrt {6} \arctan \left (\frac {\sqrt {3}+\sqrt {-3+2 x}}{2 \sqrt {2}}\right )+20 \log \left (4+x+\sqrt {-9+6 x}\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {3}{2}-x +2 \sqrt {-9+6 x}+10 \ln \left (24+6 x +6 \sqrt {-9+6 x}\right )-\frac {21 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {-9+6 x}+6\right ) \sqrt {6}}{24}\right )}{2}\) | \(54\) |
default | \(\frac {3}{2}-x +2 \sqrt {-9+6 x}+10 \ln \left (24+6 x +6 \sqrt {-9+6 x}\right )-\frac {21 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {-9+6 x}+6\right ) \sqrt {6}}{24}\right )}{2}\) | \(54\) |
trager | \(-x +2 \sqrt {-9+6 x}-\ln \left (4+x +\sqrt {-9+6 x}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+\ln \left (184 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x -10920 \sqrt {-9+6 x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )-11120 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -19635 \sqrt {-9+6 x}+19320 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+150634 x -771540\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+20 \ln \left (4+x +\sqrt {-9+6 x}\right )-10 \ln \left (184 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x -10920 \sqrt {-9+6 x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )-11120 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -19635 \sqrt {-9+6 x}+19320 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+150634 x -771540\right )\) | \(221\) |
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Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21}{2} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{12} \, \sqrt {3} \sqrt {2} \sqrt {6 \, x - 9} + \frac {1}{4} \, \sqrt {3} \sqrt {2}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (x + \sqrt {6 \, x - 9} + 4\right ) \]
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Time = 2.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=- x + 2 \sqrt {6 x - 9} + 10 \log {\left (6 x + 6 \sqrt {6 x - 9} + 24 \right )} - \frac {21 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (\sqrt {6 x - 9} + 3\right )}{12} \right )}}{2} + \frac {3}{2} \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21}{2} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) + \frac {3}{2} \]
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Time = 0.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21}{2} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (\sqrt {3} + \sqrt {2 \, x - 3}\right )}\right ) + 2 \, \sqrt {3} \sqrt {2 \, x - 3} - x + 10 \, \log \left (2 \, \sqrt {3} \sqrt {2 \, x - 3} + 2 \, x + 8\right ) + \frac {3}{2} \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=2\,\sqrt {6\,x-9}+10\,\ln \left (\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (-10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )+20\,\sqrt {6\,x-9}-66\right )\,\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )-20\,\sqrt {6\,x-9}+66\right )\right )-x-\frac {21\,\sqrt {2}\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3}}{4}+\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {6\,x-9}}{12}\right )}{2} \]
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