3.707 \(\int \frac{1+x}{4+x+\sqrt{-9+6 x}} \, dx\)

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

[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]]

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Rubi [A]  time = 0.129946, antiderivative size = 67, normalized size of antiderivative = 1., 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 = {1628, 634, 618, 204, 628} \[ x-2 \sqrt{3} \sqrt{2 x-3}+3 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )+4 \sqrt{6} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

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 1628

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]

Rule 634

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 618

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 204

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1+x}{4+x+\sqrt{-9+6 x}} \, dx &=\frac{1}{3} \operatorname{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} \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{6+2 x}{33+6 x+x^2} \, dx,x,\sqrt{-9+6 x}\right )+48 \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 0.0714985, size = 56, normalized size = 0.84 \[ x-2 \sqrt{6 x-9}+3 \log \left (x+\sqrt{6 x-9}+4\right )+4 \sqrt{6} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 52, normalized size = 0.8 \begin{align*} -2\,\sqrt{-9+6\,x}-{\frac{3}{2}}+x+3\,\ln \left ( 24+6\,x+6\,\sqrt{-9+6\,x} \right ) +4\,\sqrt{6}\arctan \left ( 1/24\, \left ( 2\,\sqrt{-9+6\,x}+6 \right ) \sqrt{6} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-9+6*x)^(1/2)-3/2+x+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 = 1.48491, size = 66, normalized size = 0.99 \begin{align*} 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{align*}

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 = 1.4617, size = 153, normalized size = 2.28 \begin{align*} 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{align*}

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 = 24.8384, size = 58, normalized size = 0.87 \begin{align*} 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{align*}

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 = 1.14779, size = 113, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, \sqrt{3} \sqrt{2}{\left (\sqrt{3} \sqrt{2} \log \left (33\right ) + 8 \, \arctan \left (\frac{1}{4} \, \sqrt{3} \sqrt{2}\right )\right )} + 4 \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{12} \, \sqrt{3} \sqrt{2}{\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{align*}

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]

-1/2*sqrt(3)*sqrt(2)*(sqrt(3)*sqrt(2)*log(33) + 8*arctan(1/4*sqrt(3)*sqrt(2))) + 4*sqrt(3)*sqrt(2)*arctan(1/12
*sqrt(3)*sqrt(2)*(sqrt(6*x - 9) + 3)) + x - 2*sqrt(6*x - 9) + 3*log(6*x + 6*sqrt(6*x - 9) + 24) - 3/2