3.448 \(\int \frac{1-\sqrt{2+3 x}}{1+\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=33 \[ -x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left (\sqrt{3 x+2}+1\right ) \]

[Out]

-x + (4*Sqrt[2 + 3*x])/3 - (4*Log[1 + Sqrt[2 + 3*x]])/3

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Rubi [A]  time = 0.0520846, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left (\sqrt{3 x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - Sqrt[2 + 3*x])/(1 + Sqrt[2 + 3*x]),x]

[Out]

-x + (4*Sqrt[2 + 3*x])/3 - (4*Log[1 + Sqrt[2 + 3*x]])/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 \sqrt{3 x + 2}}{3} - \frac{4 \log{\left (\sqrt{3 x + 2} + 1 \right )}}{3} - \frac{2 \int ^{\sqrt{3 x + 2}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-(2+3*x)**(1/2))/(1+(2+3*x)**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.0167514, size = 34, normalized size = 1.03 \[ \frac{1}{3} \left (-3 x+4 \sqrt{3 x+2}-4 \log \left (\sqrt{3 x+2}+1\right )+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Sqrt[2 + 3*x])/(1 + Sqrt[2 + 3*x]),x]

[Out]

(3 - 3*x + 4*Sqrt[2 + 3*x] - 4*Log[1 + Sqrt[2 + 3*x]])/3

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \[ -{\frac{2}{3}}-x+{\frac{4}{3}\sqrt{2+3\,x}}-{\frac{4}{3}\ln \left ( 1+\sqrt{2+3\,x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-(2+3*x)^(1/2))/(1+(2+3*x)^(1/2)),x)

[Out]

-2/3-x+4/3*(2+3*x)^(1/2)-4/3*ln(1+(2+3*x)^(1/2))

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Maxima [A]  time = 0.69277, size = 35, normalized size = 1.06 \[ -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \, \log \left (\sqrt{3 \, x + 2} + 1\right ) - \frac{2}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(3*x + 2) - 1)/(sqrt(3*x + 2) + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261773, size = 34, normalized size = 1.03 \[ -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \, \log \left (\sqrt{3 \, x + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(3*x + 2) - 1)/(sqrt(3*x + 2) + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.371301, size = 27, normalized size = 0.82 \[ - x + \frac{4 \sqrt{3 x + 2}}{3} - \frac{4 \log{\left (\sqrt{3 x + 2} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-(2+3*x)**(1/2))/(1+(2+3*x)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273814, size = 35, normalized size = 1.06 \[ -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \,{\rm ln}\left (\sqrt{3 \, x + 2} + 1\right ) - \frac{2}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(3*x + 2) - 1)/(sqrt(3*x + 2) + 1),x, algorithm="giac")

[Out]

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