\(\int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx\) [218]

Optimal result

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

[Out]

-2/9*(2+3*x)*ln(2+3*x)/(-(2+3*x)^2)^(1/2)-1/9*(-(2+3*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} \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{9} \sqrt {-9 x^2-12 x-4}-\frac {2 (3 x+2) \log (3 x+2)}{9 \sqrt {-9 x^2-12 x-4}} \]

[In]

Int[x/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

-1/9*Sqrt[-4 - 12*x - 9*x^2] - (2*(2 + 3*x)*Log[2 + 3*x])/(9*Sqrt[-4 - 12*x - 9*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}{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] (verified)

Time = 1.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {(2+3 x) (2+3 x-2 \log (2+3 x))}{9 \sqrt {-(2+3 x)^2}} \]

[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 [C] (verified)

Result contains complex when optimal does not.

Time = 2.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.33

method	result	size
meijerg	\(-\frac {2 i \left (\frac {3 x}{2}-\ln \left (1+\frac {3 x}{2}\right )\right )}{9}\)	\(16\)
default	\(-\frac {\left (2+3 x \right ) \left (-3 x +2 \ln \left (2+3 x \right )\right )}{9 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(31\)
risch	\(\frac {\left (2+3 x \right ) x}{3 \sqrt {-\left (2+3 x \right )^{2}}}-\frac {2 \left (2+3 x \right ) \ln \left (2+3 x \right )}{9 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(45\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.21 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{3} i \, x + \frac {2}{9} i \, \log \left (x + \frac {2}{3}\right ) \]

[In]

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

[Out]

-1/3*I*x + 2/9*I*log(x + 2/3)

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=- \frac {\sqrt {- 9 x^{2} - 12 x - 4}}{9} - \frac {2 \left (x + \frac {2}{3}\right ) \log {\left (x + \frac {2}{3} \right )}}{9 \sqrt {- \left (x + \frac {2}{3}\right )^{2}}} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.44 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{9} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} - \frac {2}{9} i \, \log \left (x + \frac {2}{3}\right ) \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {i \, x}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} - \frac {2 i \, \log \left ({\left | 3 \, x + 2 \right |}\right )}{9 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {\sqrt {-9\,x^2-12\,x-4}}{9}+\frac {\ln \left (x+\frac {2}{3}-\frac {\sqrt {-{\left (3\,x+2\right )}^2}\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \]

[In]

int(x/(-(3*x + 2)^2)^(1/2),x)

[Out]

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