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

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

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Rubi [A]  time = 0.14, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 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]

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 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 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 6742

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

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*}

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Mathematica [A]  time = 0.08, size = 71, normalized size = 0.90 \[ \frac {1}{576} \left (-96 x^4-64 x^3+96 \left (x^2+x+1\right )^{3/2} x-80 \left (x^2+x+1\right )^{3/2}+6 (2 x+1) \sqrt {x^2+x+1}+9 \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]

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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IntegrateAlgebraic [A]  time = 0.19, size = 67, normalized size = 0.85 \[ -\frac {1}{64} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right )+\frac {1}{18} \left (-3 x^4-2 x^3\right )+\frac {1}{288} \sqrt {x^2+x+1} \left (48 x^3+8 x^2+14 x-37\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x + x^2]*(-37 + 14*x + 8*x^2 + 48*x^3))/288 + (-2*x^3 - 3*x^4)/18 - Log[-1 - 2*x + 2*Sqrt[1 + x + x^
2]]/64

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fricas [A]  time = 0.66, size = 54, normalized size = 0.68 \[ -\frac {1}{6} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {1}{288} \, {\left (48 \, x^{3} + 8 \, x^{2} + 14 \, x - 37\right )} \sqrt {x^{2} + x + 1} - \frac {1}{64} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

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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giac [A]  time = 0.62, size = 54, normalized size = 0.68 \[ -\frac {1}{6} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {1}{288} \, {\left (2 \, {\left (4 \, {\left (6 \, x + 1\right )} x + 7\right )} x - 37\right )} \sqrt {x^{2} + x + 1} - \frac {1}{64} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

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)

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maple [A]  time = 0.07, size = 55, normalized size = 0.70

method	result	size
trager	\(-\frac {\left (2+3 x \right ) x^{3}}{18}+\frac {\left (\frac {1}{2} x^{3}+\frac {1}{12} x^{2}+\frac {7}{48} x -\frac {37}{96}\right ) \sqrt {x^{2}+x +1}}{3}-\frac {\ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )}{64}\)	\(55\)
default	\(-\frac {x^{3}}{9}-\frac {x^{4}}{6}+\frac {x \left (x^{2}+x +1\right )^{\frac {3}{2}}}{6}-\frac {5 \left (x^{2}+x +1\right )^{\frac {3}{2}}}{36}+\frac {\left (1+2 x \right ) \sqrt {x^{2}+x +1}}{96}+\frac {\arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{64}\)	\(59\)

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,method=_RETURNVERBOSE)

[Out]

-1/18*(2+3*x)*x^3+1/3*(1/2*x^3+1/12*x^2+7/48*x-37/96)*(x^2+x+1)^(1/2)-1/64*ln(2*(x^2+x+1)^(1/2)-1-2*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{2 \, x + 2 \, \sqrt {x^{2} + x + 1} + 1}\,{d x} \]

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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mupad [B]  time = 0.07, size = 71, normalized size = 0.90 \[ \frac {\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{64}-\frac {\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x^2+x+1}}{6}-\frac {x^3}{9}-\frac {x^4}{6}-\frac {5\,\left (8\,x^2+2\,x+5\right )\,\sqrt {x^2+x+1}}{288}+\frac {x\,{\left (x^2+x+1\right )}^{3/2}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (x + x^2 + 1)^(1/2) + 1/2)/64 - ((x/2 + 1/4)*(x + x^2 + 1)^(1/2))/6 - x^3/9 - x^4/6 - (5*(2*x + 8*x^2
+ 5)*(x + x^2 + 1)^(1/2))/288 + (x*(x + x^2 + 1)^(3/2))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{2 x + 2 \sqrt {x^{2} + x + 1} + 1}\, dx \]

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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