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

Optimal. Leaf size=62 \[ \frac{2 (1-x)}{3 x \sqrt{x^2+x+1}}-\frac{5 \sqrt{x^2+x+1}}{3 x}+\frac{3}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

(2*(1 - x))/(3*x*Sqrt[1 + x + x^2]) - (5*Sqrt[1 + x + x^2])/(3*x) + (3*ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])
/2

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Rubi [A]  time = 0.0266347, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {740, 806, 724, 206} \[ \frac{2 (1-x)}{3 x \sqrt{x^2+x+1}}-\frac{5 \sqrt{x^2+x+1}}{3 x}+\frac{3}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 + x + x^2)^(3/2)),x]

[Out]

(2*(1 - x))/(3*x*Sqrt[1 + x + x^2]) - (5*Sqrt[1 + x + x^2])/(3*x) + (3*ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])
/2

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 806

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

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx &=\frac{2 (1-x)}{3 x \sqrt{1+x+x^2}}+\frac{2}{3} \int \frac{\frac{5}{2}-x}{x^2 \sqrt{1+x+x^2}} \, dx\\ &=\frac{2 (1-x)}{3 x \sqrt{1+x+x^2}}-\frac{5 \sqrt{1+x+x^2}}{3 x}-\frac{3}{2} \int \frac{1}{x \sqrt{1+x+x^2}} \, dx\\ &=\frac{2 (1-x)}{3 x \sqrt{1+x+x^2}}-\frac{5 \sqrt{1+x+x^2}}{3 x}+3 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{2+x}{\sqrt{1+x+x^2}}\right )\\ &=\frac{2 (1-x)}{3 x \sqrt{1+x+x^2}}-\frac{5 \sqrt{1+x+x^2}}{3 x}+\frac{3}{2} \tanh ^{-1}\left (\frac{2+x}{2 \sqrt{1+x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0163208, size = 50, normalized size = 0.81 \[ \frac{3}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )-\frac{5 x^2+7 x+3}{3 x \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3 + 7*x + 5*x^2)/(3*x*Sqrt[1 + x + x^2]) + (3*ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])/2

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Maple [A]  time = 0.004, size = 56, normalized size = 0.9 \begin{align*} -{\frac{1}{x}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{3}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{5+10\,x}{6}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{3}{2}{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x/(x^2+x+1)^(1/2)-3/2/(x^2+x+1)^(1/2)-5/6*(1+2*x)/(x^2+x+1)^(1/2)+3/2*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2))

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Maxima [A]  time = 1.46165, size = 78, normalized size = 1.26 \begin{align*} -\frac{5 \, x}{3 \, \sqrt{x^{2} + x + 1}} - \frac{7}{3 \, \sqrt{x^{2} + x + 1}} - \frac{1}{\sqrt{x^{2} + x + 1} x} + \frac{3}{2} \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x \right |}} + \frac{2 \, \sqrt{3}}{3 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*x/sqrt(x^2 + x + 1) - 7/3/sqrt(x^2 + x + 1) - 1/(sqrt(x^2 + x + 1)*x) + 3/2*arcsinh(1/3*sqrt(3)*x/abs(x)
+ 2/3*sqrt(3)/abs(x))

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Fricas [B]  time = 2.21108, size = 258, normalized size = 4.16 \begin{align*} -\frac{10 \, x^{3} + 10 \, x^{2} - 9 \,{\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) + 9 \,{\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) + 2 \,{\left (5 \, x^{2} + 7 \, x + 3\right )} \sqrt{x^{2} + x + 1} + 10 \, x}{6 \,{\left (x^{3} + x^{2} + x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(10*x^3 + 10*x^2 - 9*(x^3 + x^2 + x)*log(-x + sqrt(x^2 + x + 1) + 1) + 9*(x^3 + x^2 + x)*log(-x + sqrt(x^
2 + x + 1) - 1) + 2*(5*x^2 + 7*x + 3)*sqrt(x^2 + x + 1) + 10*x)/(x^3 + x^2 + x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08326, size = 108, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (x + 2\right )}}{3 \, \sqrt{x^{2} + x + 1}} + \frac{x - \sqrt{x^{2} + x + 1} + 2}{{\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 1} + \frac{3}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) - \frac{3}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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