3.2498 \(\int \frac{5-x}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=57 \[ \frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}}-\frac{1}{3} \sqrt{3 x^2+5 x+2} \]

[Out]

-Sqrt[2 + 5*x + 3*x^2]/3 + (35*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/(6*Sqrt[3])

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Rubi [A]  time = 0.0425539, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}}-\frac{1}{3} \sqrt{3 x^2+5 x+2} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

-Sqrt[2 + 5*x + 3*x^2]/3 + (35*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 5.94536, size = 49, normalized size = 0.86 \[ - \frac{\sqrt{3 x^{2} + 5 x + 2}}{3} + \frac{35 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-sqrt(3*x**2 + 5*x + 2)/3 + 35*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 +
5*x + 2)))/18

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Mathematica [A]  time = 0.0238973, size = 50, normalized size = 0.88 \[ \frac{1}{18} \left (35 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )-6 \sqrt{3 x^2+5 x+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2] + 35*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]])/
18

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \[ -{\frac{1}{3}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{35\,\sqrt{3}}{18}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+5*x+2)^(1/2),x)

[Out]

-1/3*(3*x^2+5*x+2)^(1/2)+35/18*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(
1/2)

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Maxima [A]  time = 0.790642, size = 58, normalized size = 1.02 \[ \frac{35}{18} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

35/18*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 1/3*sqrt(3*x^2 +
5*x + 2)

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Fricas [A]  time = 0.271422, size = 81, normalized size = 1.42 \[ -\frac{1}{36} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} - 35 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

-1/36*sqrt(3)*(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2) - 35*log(sqrt(3)*(72*x^2 + 120*x
+ 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/sqrt(3*x**2 + 5*x + 2), x) - Integral(-5/sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.281151, size = 66, normalized size = 1.16 \[ -\frac{35}{18} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

-35/18*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/3
*sqrt(3*x^2 + 5*x + 2)