3.2405 \(\int \frac{x (1+x)^2}{(1+x+x^2)^3} \, dx\)

Optimal. Leaf size=33 \[ -\frac{(x+1) (2 x+1)}{6 \left (x^2+x+1\right )^2}-\frac{1}{6 \left (x^2+x+1\right )} \]

[Out]

-((1 + x)*(1 + 2*x))/(6*(1 + x + x^2)^2) - 1/(6*(1 + x + x^2))

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Rubi [A]  time = 0.0133338, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {818, 629} \[ -\frac{(x+1) (2 x+1)}{6 \left (x^2+x+1\right )^2}-\frac{1}{6 \left (x^2+x+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x*(1 + x)^2)/(1 + x + x^2)^3,x]

[Out]

-((1 + x)*(1 + 2*x))/(6*(1 + x + x^2)^2) - 1/(6*(1 + x + x^2))

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 629

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

Rubi steps

\begin{align*} \int \frac{x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx &=-\frac{(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac{1}{6} \int \frac{-1-2 x}{\left (1+x+x^2\right )^2} \, dx\\ &=-\frac{(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac{1}{6 \left (1+x+x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0115921, size = 22, normalized size = 0.67 \[ -\frac{3 x^2+4 x+2}{6 \left (x^2+x+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(1 + x)^2)/(1 + x + x^2)^3,x]

[Out]

-(2 + 4*x + 3*x^2)/(6*(1 + x + x^2)^2)

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Maple [A]  time = 0.005, size = 20, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ({x}^{2}+x+1 \right ) ^{2}} \left ( -{\frac{{x}^{2}}{2}}-{\frac{2\,x}{3}}-{\frac{1}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985452, size = 43, normalized size = 1.3 \begin{align*} -\frac{3 \, x^{2} + 4 \, x + 2}{6 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x^2 + 4*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)

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Fricas [A]  time = 1.18042, size = 76, normalized size = 2.3 \begin{align*} -\frac{3 \, x^{2} + 4 \, x + 2}{6 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x^2 + 4*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)

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Sympy [A]  time = 0.121625, size = 31, normalized size = 0.94 \begin{align*} - \frac{3 x^{2} + 4 x + 2}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x**2 + 4*x + 2)/(6*x**4 + 12*x**3 + 18*x**2 + 12*x + 6)

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Giac [A]  time = 1.15019, size = 27, normalized size = 0.82 \begin{align*} -\frac{3 \, x^{2} + 4 \, x + 2}{6 \,{\left (x^{2} + x + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x^2 + 4*x + 2)/(x^2 + x + 1)^2