\(\int \frac {x}{(1+x+x^2)^{3/2}} \, dx\) [269]

Optimal result

Integrand size = 12, antiderivative size = 17 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 (2+x)}{3 \sqrt {1+x+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {650} \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 (x+2)}{3 \sqrt {x^2+x+1}} \]

[In]

Int[x/(1 + x + x^2)^(3/2),x]

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (2+x)}{3 \sqrt {1+x+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 (2+x)}{3 \sqrt {1+x+x^2}} \]

[In]

Integrate[x/(1 + x + x^2)^(3/2),x]

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\)	\(14\)
trager	\(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\)	\(14\)
risch	\(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\)	\(14\)
default	\(-\frac {1}{\sqrt {x^{2}+x +1}}-\frac {1+2 x}{3 \sqrt {x^{2}+x +1}}\)	\(27\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (x^{2} + \sqrt {x^{2} + x + 1} {\left (x + 2\right )} + x + 1\right )}}{3 \, {\left (x^{2} + x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x/(x**2 + x + 1)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 \, x}{3 \, \sqrt {x^{2} + x + 1}} - \frac {4}{3 \, \sqrt {x^{2} + x + 1}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (x + 2\right )}}{3 \, \sqrt {x^{2} + x + 1}} \]

[In]

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

[Out]

-2/3*(x + 2)/sqrt(x^2 + x + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2\,x+4}{3\,\sqrt {x^2+x+1}} \]

[In]

int(x/(x + x^2 + 1)^(3/2),x)

[Out]

-(2*x + 4)/(3*(x + x^2 + 1)^(1/2))