3.8.8 \(\int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx\) [708]

Optimal. Leaf size=71 \[ -x+2 \sqrt {3} \sqrt {-3+2 x}-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {3+\sqrt {-9+6 x}}{2 \sqrt {6}}\right )+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right ) \]

[Out]

-x+10*ln(4+x+(-3+2*x)^(1/2)*3^(1/2))-21/2*arctan(1/12*(3+(-9+6*x)^(1/2))*6^(1/2))*6^(1/2)+2*(-3+2*x)^(1/2)*3^(
1/2)

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Rubi [A]
time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1642, 648, 632, 210, 642} \begin {gather*} -21 \sqrt {\frac {3}{2}} \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

-x + 2*Sqrt[3]*Sqrt[-3 + 2*x] - 21*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt[6])] + 10*Log[4 + x + Sqrt[3]
*Sqrt[-3 + 2*x]]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(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]

Rubi steps

\begin {align*} \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx &=-\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*}

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Mathematica [A]
time = 0.07, size = 67, normalized size = 0.94 \begin {gather*} \frac {1}{2} \left (3-2 x+4 \sqrt {-9+6 x}-21 \sqrt {6} \tan ^{-1}\left (\frac {\sqrt {3}+\sqrt {-3+2 x}}{2 \sqrt {2}}\right )+20 \log \left (4+x+\sqrt {-9+6 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

(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

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Maple [A]
time = 0.18, size = 54, normalized size = 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	\(-x +2 \sqrt {-9+6 x}+\RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) \ln \left (4+x +\sqrt {-9+6 x}\right )-\ln \left (184 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x +10920 \sqrt {-9+6 x}\, \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+3760 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -19320 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )-238035 \sqrt {-9+6 x}+1834 x -385140\right ) \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+10 \ln \left (184 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )^{2} x +10920 \sqrt {-9+6 x}\, \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )+3760 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right ) x -19320 \RootOf \left (8 \textit {\_Z}^{2}-160 \textit {\_Z} +2123\right )-238035 \sqrt {-9+6 x}+1834 x -385140\right )\)	\(208\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12-x)/(4+x+(-9+6*x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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))

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Maxima [A]
time = 0.50, size = 51, normalized size = 0.72 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="maxima")

[Out]

-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

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Fricas [A]
time = 0.37, size = 59, normalized size = 0.83 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="fricas")

[Out]

-21/2*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*sqrt(6*x - 9) + 1/4*sqrt(3)*sqrt(2)) - x + 2*sqrt(6*x - 9) +
10*log(x + sqrt(6*x - 9) + 4)

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Sympy [A]
time = 33.98, size = 60, normalized size = 0.85 \begin {gather*} - 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)

[Out]

-x + 2*sqrt(6*x - 9) + 10*log(6*x + 6*sqrt(6*x - 9) + 24) - 21*sqrt(6)*atan(sqrt(6)*(sqrt(6*x - 9) + 3)/12)/2
+ 3/2

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Giac [A]
time = 2.13, size = 62, normalized size = 0.87 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12-x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="giac")

[Out]

-21/2*sqrt(3)*sqrt(2)*arctan(1/4*sqrt(2)*(sqrt(3) + sqrt(2*x - 3))) + 2*sqrt(3)*sqrt(2*x - 3) - x + 10*log(2*s
qrt(3)*sqrt(2*x - 3) + 2*x + 8) + 3/2

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Mupad [B]
time = 0.03, size = 118, normalized size = 1.66 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 12)/(x + (6*x - 9)^(1/2) + 4),x)

[Out]

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)*
atan((2^(1/2)*3^(1/2))/4 + (2^(1/2)*3^(1/2)*(6*x - 9)^(1/2))/12))/2

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