3.289 \(\int \frac{x^2}{1+2 x+2 \sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{x^4}{6}-\frac{x^3}{9}+\frac{1}{6} \left (x^2+x+1\right )^{3/2} x-\frac{5}{36} \left (x^2+x+1\right )^{3/2}+\frac{1}{96} (2 x+1) \sqrt{x^2+x+1}+\frac{1}{64} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

-x^3/9 - x^4/6 + ((1 + 2*x)*Sqrt[1 + x + x^2])/96 - (5*(1 + x + x^2)^(3/2))/36 + (x*(1 + x + x^2)^(3/2))/6 + A
rcSinh[(1 + 2*x)/Sqrt[3]]/64

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Rubi [A]  time = 0.135887, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6742, 742, 640, 612, 619, 215} \[ -\frac{x^4}{6}-\frac{x^3}{9}+\frac{1}{6} \left (x^2+x+1\right )^{3/2} x-\frac{5}{36} \left (x^2+x+1\right )^{3/2}+\frac{1}{96} (2 x+1) \sqrt{x^2+x+1}+\frac{1}{64} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 + 2*x + 2*Sqrt[1 + x + x^2]),x]

[Out]

-x^3/9 - x^4/6 + ((1 + 2*x)*Sqrt[1 + x + x^2])/96 - (5*(1 + x + x^2)^(3/2))/36 + (x*(1 + x + x^2)^(3/2))/6 + A
rcSinh[(1 + 2*x)/Sqrt[3]]/64

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 742

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

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 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^2}{1+2 x+2 \sqrt{1+x+x^2}} \, dx &=\int \left (-\frac{x^2}{3}-\frac{2 x^3}{3}+\frac{2}{3} x^2 \sqrt{1+x+x^2}\right ) \, dx\\ &=-\frac{x^3}{9}-\frac{x^4}{6}+\frac{2}{3} \int x^2 \sqrt{1+x+x^2} \, dx\\ &=-\frac{x^3}{9}-\frac{x^4}{6}+\frac{1}{6} x \left (1+x+x^2\right )^{3/2}+\frac{1}{6} \int \left (-1-\frac{5 x}{2}\right ) \sqrt{1+x+x^2} \, dx\\ &=-\frac{x^3}{9}-\frac{x^4}{6}-\frac{5}{36} \left (1+x+x^2\right )^{3/2}+\frac{1}{6} x \left (1+x+x^2\right )^{3/2}+\frac{1}{24} \int \sqrt{1+x+x^2} \, dx\\ &=-\frac{x^3}{9}-\frac{x^4}{6}+\frac{1}{96} (1+2 x) \sqrt{1+x+x^2}-\frac{5}{36} \left (1+x+x^2\right )^{3/2}+\frac{1}{6} x \left (1+x+x^2\right )^{3/2}+\frac{1}{64} \int \frac{1}{\sqrt{1+x+x^2}} \, dx\\ &=-\frac{x^3}{9}-\frac{x^4}{6}+\frac{1}{96} (1+2 x) \sqrt{1+x+x^2}-\frac{5}{36} \left (1+x+x^2\right )^{3/2}+\frac{1}{6} x \left (1+x+x^2\right )^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,1+2 x\right )}{64 \sqrt{3}}\\ &=-\frac{x^3}{9}-\frac{x^4}{6}+\frac{1}{96} (1+2 x) \sqrt{1+x+x^2}-\frac{5}{36} \left (1+x+x^2\right )^{3/2}+\frac{1}{6} x \left (1+x+x^2\right )^{3/2}+\frac{1}{64} \sinh ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0742065, size = 71, normalized size = 0.9 \[ \frac{1}{576} \left (-96 x^4-64 x^3+96 \left (x^2+x+1\right )^{3/2} x-80 \left (x^2+x+1\right )^{3/2}+6 (2 x+1) \sqrt{x^2+x+1}+9 \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*x^3 - 96*x^4 + 6*(1 + 2*x)*Sqrt[1 + x + x^2] - 80*(1 + x + x^2)^(3/2) + 96*x*(1 + x + x^2)^(3/2) + 9*ArcS
inh[(1 + 2*x)/Sqrt[3]])/576

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Maple [A]  time = 0.003, size = 59, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{9}}-{\frac{{x}^{4}}{6}}+{\frac{x}{6} \left ({x}^{2}+x+1 \right ) ^{{\frac{3}{2}}}}-{\frac{5}{36} \left ({x}^{2}+x+1 \right ) ^{{\frac{3}{2}}}}+{\frac{1+2\,x}{96}\sqrt{{x}^{2}+x+1}}+{\frac{1}{64}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*x^3-1/6*x^4+1/6*x*(x^2+x+1)^(3/2)-5/36*(x^2+x+1)^(3/2)+1/96*(1+2*x)*(x^2+x+1)^(1/2)+1/64*arcsinh(2/3*3^(1
/2)*(x+1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{2 \, x + 2 \, \sqrt{x^{2} + x + 1} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(2*x + 2*sqrt(x^2 + x + 1) + 1), x)

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Fricas [A]  time = 1.86522, size = 159, normalized size = 2.01 \begin{align*} -\frac{1}{6} \, x^{4} - \frac{1}{9} \, x^{3} + \frac{1}{288} \,{\left (48 \, x^{3} + 8 \, x^{2} + 14 \, x - 37\right )} \sqrt{x^{2} + x + 1} - \frac{1}{64} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x^4 - 1/9*x^3 + 1/288*(48*x^3 + 8*x^2 + 14*x - 37)*sqrt(x^2 + x + 1) - 1/64*log(-2*x + 2*sqrt(x^2 + x + 1
) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06957, size = 73, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, x^{4} - \frac{1}{9} \, x^{3} + \frac{1}{288} \,{\left (2 \,{\left (4 \,{\left (6 \, x + 1\right )} x + 7\right )} x - 37\right )} \sqrt{x^{2} + x + 1} - \frac{1}{64} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x^4 - 1/9*x^3 + 1/288*(2*(4*(6*x + 1)*x + 7)*x - 37)*sqrt(x^2 + x + 1) - 1/64*log(-2*x + 2*sqrt(x^2 + x +
1) - 1)