\(\int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx\) [2432]

Optimal result

Integrand size = 18, antiderivative size = 36 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {4+5 x}{2 \sqrt {2} \sqrt {2+5 x+3 x^2}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 212} \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {5 x+4}{2 \sqrt {2} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {2}} \]

[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])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+5 x}{\sqrt {2+5 x+3 x^2}}\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {4+5 x}{2 \sqrt {2} \sqrt {2+5 x+3 x^2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2+5 x+3 x^2}}{\sqrt {2} (1+x)}\right ) \]

[In]

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

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[2]*(1 + x))])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\operatorname {arctanh}\left (\frac {\left (4+5 x \right ) \sqrt {2}}{4 \sqrt {3 x^{2}+5 x +2}}\right ) \sqrt {2}}{2}\)	\(29\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-4 \sqrt {3 x^{2}+5 x +2}}{x}\right )}{2}\)	\(47\)

[In]

int(1/x/(3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, \sqrt {2} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x + 4\right )} - 49 \, x^{2} - 80 \, x - 32}{x^{2}}\right ) \]

[In]

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

\[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=\int \frac {1}{x \sqrt {\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 5\right ) \]

[In]

integrate(1/x/(3*x^2+5*x+2)^(1/2),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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {3} x + \sqrt {2} + \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {3} x - \sqrt {2} + \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \sqrt {2+5 x+3 x^2}} \, dx=-\frac {\sqrt {2}\,\ln \left (\frac {5\,x+2\,\sqrt {6\,x^2+10\,x+4}+4}{x}\right )}{2} \]

[In]

int(1/(x*(5*x + 3*x^2 + 2)^(1/2)),x)

[Out]

-(2^(1/2)*log((5*x + 2*(10*x + 6*x^2 + 4)^(1/2) + 4)/x))/2