\(\int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx\) [288]

Optimal result

Integrand size = 14, antiderivative size = 25 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {2+3 x}{\sqrt {-5+12 x+9 x^2}}\right ) \]

[Out]

1/3*arctanh((2+3*x)/(9*x^2+12*x-5)^(1/2))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {635, 212} \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {3 x+2}{\sqrt {9 x^2+12 x-5}}\right ) \]

[In]

Int[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

ArcTanh[(2 + 3*x)/Sqrt[-5 + 12*x + 9*x^2]]/3

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 635

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {12+18 x}{\sqrt {-5+12 x+9 x^2}}\right ) \\ & = \frac {1}{3} \text {arctanh}\left (\frac {2+3 x}{\sqrt {-5+12 x+9 x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=-\frac {1}{3} \log \left (-2-3 x+\sqrt {-5+12 x+9 x^2}\right ) \]

[In]

Integrate[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

-1/3*Log[-2 - 3*x + Sqrt[-5 + 12*x + 9*x^2]]

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

method	result	size
trager	\(-\frac {\ln \left (-2-3 x +\sqrt {9 x^{2}+12 x -5}\right )}{3}\)	\(21\)
default	\(\frac {\ln \left (\frac {\left (9 x +6\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+12 x -5}\right ) \sqrt {9}}{9}\)	\(30\)

[In]

int(1/(9*x^2+12*x-5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(-2-3*x+(9*x^2+12*x-5)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=-\frac {1}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x - 5} - 2\right ) \]

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="fricas")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 + 12*x - 5) - 2)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {\log {\left (18 x + 6 \sqrt {9 x^{2} + 12 x - 5} + 12 \right )}}{3} \]

[In]

integrate(1/(9*x**2+12*x-5)**(1/2),x)

[Out]

log(18*x + 6*sqrt(9*x**2 + 12*x - 5) + 12)/3

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {1}{3} \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} + 12 \, x - 5} + 12\right ) \]

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="maxima")

[Out]

1/3*log(18*x + 6*sqrt(9*x^2 + 12*x - 5) + 12)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {1}{6} \, \sqrt {9 \, x^{2} + 12 \, x - 5} {\left (3 \, x + 2\right )} + \frac {3}{2} \, \log \left ({\left | -3 \, x + \sqrt {9 \, x^{2} + 12 \, x - 5} - 2 \right |}\right ) \]

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(9*x^2 + 12*x - 5)*(3*x + 2) + 3/2*log(abs(-3*x + sqrt(9*x^2 + 12*x - 5) - 2))

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx=\frac {\ln \left (3\,x+\sqrt {9\,x^2+12\,x-5}+2\right )}{3} \]

[In]

int(1/(12*x + 9*x^2 - 5)^(1/2),x)

[Out]

log(3*x + (12*x + 9*x^2 - 5)^(1/2) + 2)/3