Integrand size = 20, antiderivative size = 71 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-x+2 \sqrt {3} \sqrt {-3+2 x}-21 \sqrt {\frac {3}{2}} \arctan \left (\frac {3+\sqrt {-9+6 x}}{2 \sqrt {6}}\right )+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right ) \] Output:
-x+2*(-3+2*x)^(1/2)*3^(1/2)-21/2*6^(1/2)*arctan(1/12*(3+(-9+6*x)^(1/2))*6^ (1/2))+10*ln(4+x+(-3+2*x)^(1/2)*3^(1/2))
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=\frac {1}{2} \left (3-2 x+4 \sqrt {-9+6 x}-21 \sqrt {6} \arctan \left (\frac {\sqrt {3}+\sqrt {-3+2 x}}{2 \sqrt {2}}\right )+20 \log \left (4+x+\sqrt {-9+6 x}\right )\right ) \] Input:
Integrate[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]
Output:
(3 - 2*x + 4*Sqrt[-9 + 6*x] - 21*Sqrt[6]*ArcTan[(Sqrt[3] + Sqrt[-3 + 2*x]) /(2*Sqrt[2])] + 20*Log[4 + x + Sqrt[-9 + 6*x]])/2
Time = 0.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7267, 25, 2159, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {12-x}{x+\sqrt {6 x-9}+4} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle -\frac {1}{3} \int -\frac {(72-6 x) \sqrt {6 x-9}}{6 x+6 \sqrt {6 x-9}+24}d\sqrt {6 x-9}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \int \frac {(72-6 x) \sqrt {6 x-9}}{6 x+6 \sqrt {6 x-9}+24}d\sqrt {6 x-9}\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \frac {1}{3} \int \left (-\frac {6 \left (33-10 \sqrt {6 x-9}\right )}{6 x+6 \sqrt {6 x-9}+24}-\sqrt {6 x-9}+6\right )d\sqrt {6 x-9}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-63 \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right )+\frac {1}{2} (9-6 x)+6 \sqrt {6 x-9}+30 \log \left (6 x+6 \sqrt {6 x-9}+24\right )\right )\) |
Input:
Int[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]
Output:
((9 - 6*x)/2 + 6*Sqrt[-9 + 6*x] - 63*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x]) /(2*Sqrt[6])] + 30*Log[24 + 6*x + 6*Sqrt[-9 + 6*x]])/3
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.15 (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 | \(-x +2 \sqrt {-9+6 x}+\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) \ln \left (4+x +\sqrt {-9+6 x}\right )-\ln \left (184 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x +10920 \sqrt {-9+6 x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+3760 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -238035 \sqrt {-9+6 x}-19320 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+1834 x -385140\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+10 \ln \left (184 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x +10920 \sqrt {-9+6 x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+3760 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -238035 \sqrt {-9+6 x}-19320 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+1834 x -385140\right )\) | \(208\) |
Input:
int((12-x)/(4+x+(-9+6*x)^(1/2)),x,method=_RETURNVERBOSE)
Output:
3/2-x+2*(-9+6*x)^(1/2)+10*ln(24+6*x+6*(-9+6*x)^(1/2))-21/2*6^(1/2)*arctan( 1/24*(2*(-9+6*x)^(1/2)+6)*6^(1/2))
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.70 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-21 \, \sqrt {\frac {3}{2}} \arctan \left (\frac {1}{6} \, \sqrt {\frac {3}{2}} \sqrt {6 \, x - 9} + \frac {1}{2} \, \sqrt {\frac {3}{2}}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (x + \sqrt {6 \, x - 9} + 4\right ) \] Input:
integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="fricas")
Output:
-21*sqrt(3/2)*arctan(1/6*sqrt(3/2)*sqrt(6*x - 9) + 1/2*sqrt(3/2)) - x + 2* sqrt(6*x - 9) + 10*log(x + sqrt(6*x - 9) + 4)
Time = 2.52 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=- x + 2 \sqrt {6 x - 9} + 10 \log {\left (6 x + 6 \sqrt {6 x - 9} + 24 \right )} - \frac {21 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (\sqrt {6 x - 9} + 3\right )}{12} \right )}}{2} + \frac {3}{2} \] Input:
integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)
Output:
-x + 2*sqrt(6*x - 9) + 10*log(6*x + 6*sqrt(6*x - 9) + 24) - 21*sqrt(6)*ata n(sqrt(6)*(sqrt(6*x - 9) + 3)/12)/2 + 3/2
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21}{2} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) + \frac {3}{2} \] Input:
integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="maxima")
Output:
-21/2*sqrt(6)*arctan(1/12*sqrt(6)*(sqrt(6*x - 9) + 3)) - x + 2*sqrt(6*x - 9) + 10*log(6*x + 6*sqrt(6*x - 9) + 24) + 3/2
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21}{2} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (\sqrt {3} + \sqrt {2 \, x - 3}\right )}\right ) + 2 \, \sqrt {3} \sqrt {2 \, x - 3} - x + 10 \, \log \left (2 \, \sqrt {3} \sqrt {2 \, x - 3} + 2 \, x + 8\right ) + \frac {3}{2} \] Input:
integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="giac")
Output:
-21/2*sqrt(3)*sqrt(2)*arctan(1/4*sqrt(2)*(sqrt(3) + sqrt(2*x - 3))) + 2*sq rt(3)*sqrt(2*x - 3) - x + 10*log(2*sqrt(3)*sqrt(2*x - 3) + 2*x + 8) + 3/2
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=2\,\sqrt {6\,x-9}+10\,\ln \left (\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (-10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )+20\,\sqrt {6\,x-9}-66\right )\,\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )-20\,\sqrt {6\,x-9}+66\right )\right )-x-\frac {21\,\sqrt {2}\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3}}{4}+\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {6\,x-9}}{12}\right )}{2} \] Input:
int(-(x - 12)/(x + (6*x - 9)^(1/2) + 4),x)
Output:
10*log(((2*(6*x - 9)^(1/2) + 6)*((2^(1/2)*3^(1/2)*21i)/4 - 10) + 20*(6*x - 9)^(1/2) - 66)*((2*(6*x - 9)^(1/2) + 6)*((2^(1/2)*3^(1/2)*21i)/4 + 10) - 20*(6*x - 9)^(1/2) + 66)) - x + 2*(6*x - 9)^(1/2) - (21*2^(1/2)*3^(1/2)*at an((2^(1/2)*3^(1/2))/4 + (2^(1/2)*3^(1/2)*(6*x - 9)^(1/2))/12))/2
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx=-\frac {21 \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {2 x -3}+\sqrt {3}}{2 \sqrt {2}}\right )}{2}+2 \sqrt {2 x -3}\, \sqrt {3}+10 \,\mathrm {log}\left (2 \sqrt {2 x -3}\, \sqrt {3}+2 x +8\right )-x +\frac {3}{2} \] Input:
int((12-x)/(4+x+(-9+6*x)^(1/2)),x)
Output:
( - 21*sqrt(6)*atan((sqrt(2*x - 3) + sqrt(3))/(2*sqrt(2))) + 4*sqrt(2*x - 3)*sqrt(3) + 20*log(2*sqrt(2*x - 3)*sqrt(3) + 2*x + 8) - 2*x + 3)/2