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

Optimal. Leaf size=48 \[ \frac{1}{9} \sqrt{9 x^2+12 x+4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{9 x^2+12 x+4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0113589, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 608, 31} \[ \frac{1}{9} \sqrt{9 x^2+12 x+4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

Rule 640

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]

Rule 608

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 31

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

Rubi steps

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

Mathematica [A]  time = 0.0110616, size = 33, normalized size = 0.69 \[ \frac{(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt{(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.12, size = 29, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2+3\,x \right ) \left ( -3\,x+2\,\ln \left ( 2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{ \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((2+3*x)^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.67392, size = 28, normalized size = 0.58 \begin{align*} \frac{1}{9} \, \sqrt{9 \, x^{2} + 12 \, x + 4} - \frac{2}{9} \, \log \left (x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.68502, size = 35, normalized size = 0.73 \begin{align*} \frac{1}{3} \, x - \frac{2}{9} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x - 2/9*log(3*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.173751, size = 12, normalized size = 0.25 \begin{align*} \frac{x}{3} - \frac{2 \log{\left (3 x + 2 \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/3 - 2*log(3*x + 2)/9

________________________________________________________________________________________

Giac [A]  time = 1.40761, size = 34, normalized size = 0.71 \begin{align*} \frac{1}{3} \, x \mathrm{sgn}\left (3 \, x + 2\right ) - \frac{2}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm{sgn}\left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x*sgn(3*x + 2) - 2/9*log(abs(3*x + 2))*sgn(3*x + 2)