3.8.7 \(\int \frac {1+x}{4+x+\sqrt {-9+6 x}} \, dx\) [707]

Optimal. Leaf size=67 \[ x-2 \sqrt {3} \sqrt {-3+2 x}+4 \sqrt {6} \tan ^{-1}\left (\frac {3+\sqrt {-9+6 x}}{2 \sqrt {6}}\right )+3 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right ) \]

[Out]

x+3*ln(4+x+(-3+2*x)^(1/2)*3^(1/2))+4*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 = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1642, 648, 632, 210, 642} \begin {gather*} 4 \sqrt {6} \text {ArcTan}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right )+x-2 \sqrt {3} \sqrt {2 x-3}+3 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

x - 2*Sqrt[3]*Sqrt[-3 + 2*x] + 4*Sqrt[6]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt[6])] + 3*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 {1+x}{4+x+\sqrt {-9+6 x}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x \left (15+x^2\right )}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-6+x+\frac {18 (11+x)}{33+6 x+x^2}\right ) \, dx,x,\sqrt {-9+6 x}\right )\\ &=x-2 \sqrt {3} \sqrt {-3+2 x}+6 \text {Subst}\left (\int \frac {11+x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=x-2 \sqrt {3} \sqrt {-3+2 x}+3 \text {Subst}\left (\int \frac {6+2 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )+48 \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}+3 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )-96 \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}+4 \sqrt {6} \tan ^{-1}\left (\frac {3+\sqrt {3} \sqrt {-3+2 x}}{2 \sqrt {6}}\right )+3 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 63, normalized size = 0.94 \begin {gather*} -\frac {3}{2}+x-2 \sqrt {-9+6 x}+4 \sqrt {6} \tan ^{-1}\left (\frac {\sqrt {3}+\sqrt {-3+2 x}}{2 \sqrt {2}}\right )+3 \log \left (4+x+\sqrt {-9+6 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

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

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Maple [A]
time = 0.19, size = 52, normalized size = 0.78

method	result	size
derivativedivides	\(-\frac {3}{2}+x -2 \sqrt {-9+6 x}+3 \ln \left (24+6 x +6 \sqrt {-9+6 x}\right )+4 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {-9+6 x}+6\right ) \sqrt {6}}{24}\right )\)	\(52\)
default	\(-\frac {3}{2}+x -2 \sqrt {-9+6 x}+3 \ln \left (24+6 x +6 \sqrt {-9+6 x}\right )+4 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {-9+6 x}+6\right ) \sqrt {6}}{24}\right )\)	\(52\)
trager	\(x -2 \sqrt {-9+6 x}+\RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right ) \ln \left (4+x +\sqrt {-9+6 x}\right )-\ln \left (161 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )^{2} x -3640 \sqrt {-9+6 x}\, \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )-3446 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right ) x +6440 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )-5440 \sqrt {-9+6 x}+16233 x -92760\right ) \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )+3 \ln \left (161 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )^{2} x -3640 \sqrt {-9+6 x}\, \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )-3446 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right ) x +6440 \RootOf \left (\textit {\_Z}^{2}-6 \textit {\_Z} +33\right )-5440 \sqrt {-9+6 x}+16233 x -92760\right )\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2+x-2*(-9+6*x)^(1/2)+3*ln(24+6*x+6*(-9+6*x)^(1/2))+4*6^(1/2)*arctan(1/24*(2*(-9+6*x)^(1/2)+6)*6^(1/2))

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Maxima [A]
time = 0.49, size = 49, normalized size = 0.73 \begin {gather*} 4 \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt {6 \, x - 9} + 3 \, \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((1+x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="maxima")

[Out]

4*sqrt(6)*arctan(1/12*sqrt(6)*(sqrt(6*x - 9) + 3)) + x - 2*sqrt(6*x - 9) + 3*log(6*x + 6*sqrt(6*x - 9) + 24) -
3/2

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Fricas [A]
time = 0.34, size = 48, normalized size = 0.72 \begin {gather*} 4 \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} \sqrt {6 \, x - 9} + \frac {1}{4} \, \sqrt {6}\right ) + x - 2 \, \sqrt {6 \, x - 9} + 3 \, \log \left (x + \sqrt {6 \, x - 9} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 18.19, size = 58, normalized size = 0.87 \begin {gather*} x - 2 \sqrt {6 x - 9} + 3 \log {\left (6 x + 6 \sqrt {6 x - 9} + 24 \right )} + 4 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (\sqrt {6 x - 9} + 3\right )}{12} \right )} - \frac {3}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*sqrt(6*x - 9) + 3*log(6*x + 6*sqrt(6*x - 9) + 24) + 4*sqrt(6)*atan(sqrt(6)*(sqrt(6*x - 9) + 3)/12) - 3/2

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Giac [A]
time = 2.42, size = 60, normalized size = 0.90 \begin {gather*} 4 \, \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 + 3 \, \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((1+x)/(4+x+(-9+6*x)^(1/2)),x, algorithm="giac")

[Out]

4*sqrt(3)*sqrt(2)*arctan(1/4*sqrt(2)*(sqrt(3) + sqrt(2*x - 3))) - 2*sqrt(3)*sqrt(2*x - 3) + x + 3*log(2*sqrt(3
)*sqrt(2*x - 3) + 2*x + 8) - 3/2

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Mupad [B]
time = 3.09, size = 102, normalized size = 1.52 \begin {gather*} x+3\,\ln \left (\left (6\,\sqrt {6\,x-9}+\left (-3+\sqrt {6}\,2{}\mathrm {i}\right )\,\left (2\,\sqrt {6\,x-9}+6\right )+66\right )\,\left (6\,\sqrt {6\,x-9}-\left (3+\sqrt {6}\,2{}\mathrm {i}\right )\,\left (2\,\sqrt {6\,x-9}+6\right )+66\right )\right )+4\,\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,\sqrt {6\,x-9}}{12}+\frac {\sqrt {6}}{4}\right )-2\,\sqrt {6\,x-9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*log((6*(6*x - 9)^(1/2) + (6^(1/2)*2i - 3)*(2*(6*x - 9)^(1/2) + 6) + 66)*(6*(6*x - 9)^(1/2) - (6^(1/2)*2i
+ 3)*(2*(6*x - 9)^(1/2) + 6) + 66)) + 4*6^(1/2)*atan((6^(1/2)*(6*x - 9)^(1/2))/12 + 6^(1/2)/4) - 2*(6*x - 9)^
(1/2)

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