3.2497 \(\int \frac{(5-x) (3+2 x)}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{6} \sqrt{3 x^2+5 x+2} (19-2 x)+\frac{31 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4 \sqrt{3}} \]

[Out]

((19 - 2*x)*Sqrt[2 + 5*x + 3*x^2])/6 + (31*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(4*Sqrt[3])

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Rubi [A]  time = 0.067815, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{1}{6} \sqrt{3 x^2+5 x+2} (19-2 x)+\frac{31 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((19 - 2*x)*Sqrt[2 + 5*x + 3*x^2])/6 + (31*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(4*Sqrt[3])

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Rubi in Sympy [A]  time = 9.16686, size = 54, normalized size = 0.87 \[ \frac{\left (- 6 x + 57\right ) \sqrt{3 x^{2} + 5 x + 2}}{18} + \frac{31 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-6*x + 57)*sqrt(3*x**2 + 5*x + 2)/18 + 31*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sq
rt(3*x**2 + 5*x + 2)))/12

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Mathematica [A]  time = 0.0530775, size = 55, normalized size = 0.89 \[ \frac{31 \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )}{4 \sqrt{3}}-\frac{1}{6} (2 x-19) \sqrt{3 x^2+5 x+2} \]

Antiderivative was successfully verified.

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

[Out]

-((-19 + 2*x)*Sqrt[2 + 5*x + 3*x^2])/6 + (31*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x
^2]])/(4*Sqrt[3])

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Maple [A]  time = 0.007, size = 60, normalized size = 1. \[{\frac{31\,\sqrt{3}}{12}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{19}{6}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{x}{3}\sqrt{3\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

31/12*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+19/6*(3*x^2+5*x+2)^(
1/2)-1/3*x*(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.787794, size = 78, normalized size = 1.26 \[ -\frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{31}{12} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{19}{6} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3*x^2 + 5*x + 2)*x + 31/12*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2)
+ 6*x + 5) + 19/6*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.272445, size = 88, normalized size = 1.42 \[ -\frac{1}{72} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x - 19\right )} - 93 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*sqrt(3)*(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(2*x - 19) - 93*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. \[ - \int \left (- \frac{7 x}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{2 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{15}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-7*x/sqrt(3*x**2 + 5*x + 2), x) - Integral(2*x**2/sqrt(3*x**2 + 5*x +
2), x) - Integral(-15/sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.26996, size = 73, normalized size = 1.18 \[ -\frac{1}{6} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x - 19\right )} - \frac{31}{12} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3*x^2 + 5*x + 2)*(2*x - 19) - 31/12*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2)) - 5))