3.2421 \(\int \frac{1}{x \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.036912, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.17256, size = 34, normalized size = 0.94 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (5 x + 4\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*(5*x + 4)/(4*sqrt(3*x**2 + 5*x + 2)))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0346353, size = 33, normalized size = 0.92 \[ \frac{\log (x)-\log \left (2 \sqrt{6 x^2+10 x+4}+5 x+4\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(Log[x] - Log[4 + 5*x + 2*Sqrt[4 + 10*x + 6*x^2]])/Sqrt[2]

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 29, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+5\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^2+5*x+2)^(1/2),x)

[Out]

-1/2*arctanh(1/4*(4+5*x)*2^(1/2)/(3*x^2+5*x+2)^(1/2))*2^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.749242, size = 47, normalized size = 1.31 \[ -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*log(2*sqrt(2)*sqrt(3*x^2 + 5*x + 2)/abs(x) + 4/abs(x) + 5)

_______________________________________________________________________________________

Fricas [A]  time = 0.221251, size = 58, normalized size = 1.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{4 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x + 4\right )} - 49 \, x^{2} - 80 \, x - 32}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(-(4*sqrt(2)*sqrt(3*x^2 + 5*x + 2)*(5*x + 4) - 49*x^2 - 80*x - 32
)/x^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.233263, size = 82, normalized size = 2.28 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{3} x + \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{3} x - \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="giac")

[Out]

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