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

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 = 0.220887, 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 \[ x-2 \sqrt{3} \sqrt{2 x-3}+3 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )+4 \sqrt{6} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

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]]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 2 \sqrt{3} \sqrt{2 x - 3} + 3 \log{\left (2 x + 2 \sqrt{3} \sqrt{2 x - 3} + 8 \right )} + 4 \sqrt{6} \operatorname{atan}{\left (\sqrt{2} \left (\frac{\sqrt{2 x - 3}}{4} + \frac{\sqrt{3}}{4}\right ) \right )} + \int ^{\sqrt{2 x - 3}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(4+x+(-9+6*x)**(1/2)),x)

[Out]

-2*sqrt(3)*sqrt(2*x - 3) + 3*log(2*x + 2*sqrt(3)*sqrt(2*x - 3) + 8) + 4*sqrt(6)*
atan(sqrt(2)*(sqrt(2*x - 3)/4 + sqrt(3)/4)) + Integral(x, (x, sqrt(2*x - 3)))

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Mathematica [A]  time = 0.0721075, size = 64, normalized size = 0.96 \[ x-2 \sqrt{6 x-9}+3 \log \left (6 x+6 \left (\sqrt{6 x-9}+4\right )\right )+4 \sqrt{6} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right )-\frac{3}{2} \]

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

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Maple [A]  time = 0.009, size = 52, normalized size = 0.8 \[ -2\,\sqrt{-9+6\,x}-{\frac{3}{2}}+x+3\,\ln \left ( 24+6\,x+6\,\sqrt{-9+6\,x} \right ) +4\,\sqrt{6}\arctan \left ( 1/24\, \left ( 6+2\,\sqrt{-9+6\,x} \right ) \sqrt{6} \right ) \]

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

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Maxima [A]  time = 0.806309, size = 66, normalized size = 0.99 \[ 4 \, \sqrt{6} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt{6 \, x - 9} + 3 \, \log \left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) - \frac{3}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x + sqrt(6*x - 9) + 4),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 = 0.274862, size = 59, normalized size = 0.88 \[ 4 \, \sqrt{6} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt{6 \, x - 9} + 3 \, \log \left (x + \sqrt{6 \, x - 9} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x + sqrt(6*x - 9) + 4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{x + \sqrt{3} \sqrt{2 x - 3} + 4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)/(x + sqrt(3)*sqrt(2*x - 3) + 4), x)

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GIAC/XCAS [A]  time = 0.283529, size = 113, normalized size = 1.69 \[ -\frac{1}{2} \, \sqrt{3} \sqrt{2}{\left (\sqrt{3} \sqrt{2}{\rm ln}\left (33\right ) + 8 \, \arctan \left (\frac{1}{4} \, \sqrt{3} \sqrt{2}\right )\right )} + 4 \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{12} \, \sqrt{3} \sqrt{2}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt{6 \, x - 9} + 3 \,{\rm ln}\left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) - \frac{3}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x + sqrt(6*x - 9) + 4),x, algorithm="giac")

[Out]

-1/2*sqrt(3)*sqrt(2)*(sqrt(3)*sqrt(2)*ln(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*ln(6*x + 6*sqrt(6*x - 9) + 24) - 3/2