\(\int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx\) [211]

Optimal result

Integrand size = 16, antiderivative size = 48 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {3 (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 622, 31} \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{4} \sqrt {4 x^2+12 x+9}-\frac {3 (2 x+3) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}} \]

[In]

Int[x/Sqrt[9 + 12*x + 4*x^2],x]

[Out]

Sqrt[9 + 12*x + 4*x^2]/4 - (3*(3 + 2*x)*Log[3 + 2*x])/(4*Sqrt[9 + 12*x + 4*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 622

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

Rule 654

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {3}{2} \int \frac {1}{\sqrt {9+12 x+4 x^2}} \, dx \\ & = \frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {(3 (6+4 x)) \int \frac {1}{6+4 x} \, dx}{2 \sqrt {9+12 x+4 x^2}} \\ & = \frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {3 (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {(3+2 x) \left (\frac {x}{2}-\frac {3}{4} \log (3+2 x)\right )}{\sqrt {(3+2 x)^2}} \]

[In]

Integrate[x/Sqrt[9 + 12*x + 4*x^2],x]

[Out]

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

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.27

method	result	size
meijerg	\(\frac {x}{2}-\frac {3 \ln \left (1+\frac {2 x}{3}\right )}{4}\)	\(13\)
default	\(-\frac {\left (2 x +3\right ) \left (-2 x +3 \ln \left (2 x +3\right )\right )}{4 \sqrt {\left (2 x +3\right )^{2}}}\)	\(29\)
risch	\(\frac {x \sqrt {\left (2 x +3\right )^{2}}}{4 x +6}-\frac {3 \sqrt {\left (2 x +3\right )^{2}}\, \ln \left (2 x +3\right )}{4 \left (2 x +3\right )}\)	\(45\)

[In]

int(x/(4*x^2+12*x+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x-3/4*ln(1+2/3*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.25 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{2} \, x - \frac {3}{4} \, \log \left (2 \, x + 3\right ) \]

[In]

integrate(x/(4*x^2+12*x+9)^(1/2),x, algorithm="fricas")

[Out]

1/2*x - 3/4*log(2*x + 3)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=- \frac {3 \left (x + \frac {3}{2}\right ) \log {\left (x + \frac {3}{2} \right )}}{4 \sqrt {\left (x + \frac {3}{2}\right )^{2}}} + \frac {\sqrt {4 x^{2} + 12 x + 9}}{4} \]

[In]

integrate(x/(4*x**2+12*x+9)**(1/2),x)

[Out]

-3*(x + 3/2)*log(x + 3/2)/(4*sqrt((x + 3/2)**2)) + sqrt(4*x**2 + 12*x + 9)/4

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.44 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} - \frac {3}{4} \, \log \left (x + \frac {3}{2}\right ) \]

[In]

integrate(x/(4*x^2+12*x+9)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(4*x^2 + 12*x + 9) - 3/4*log(x + 3/2)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {x}{2 \, \mathrm {sgn}\left (2 \, x + 3\right )} - \frac {3 \, \log \left ({\left | 2 \, x + 3 \right |}\right )}{4 \, \mathrm {sgn}\left (2 \, x + 3\right )} \]

[In]

integrate(x/(4*x^2+12*x+9)^(1/2),x, algorithm="giac")

[Out]

1/2*x/sgn(2*x + 3) - 3/4*log(abs(2*x + 3))/sgn(2*x + 3)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx=\frac {\sqrt {4\,x^2+12\,x+9}}{4}-\frac {3\,\ln \left (x+\frac {\sqrt {{\left (2\,x+3\right )}^2}}{2}+\frac {3}{2}\right )}{4} \]

[In]

int(x/(12*x + 4*x^2 + 9)^(1/2),x)

[Out]

(12*x + 4*x^2 + 9)^(1/2)/4 - (3*log(x + ((2*x + 3)^2)^(1/2)/2 + 3/2))/4