3.283 \(\int \frac {1+x^4}{(1+x+x^2) \sqrt {2+x+x^2}} \, dx\)

Optimal. Leaf size=87 \[ \frac {1}{2} \sqrt {x^2+x+2} x-\frac {7}{4} \sqrt {x^2+x+2}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+x+2}}\right )}{\sqrt {3}}-\tanh ^{-1}\left (\sqrt {x^2+x+2}\right )-\frac {1}{8} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right ) \]

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Rubi [A]  time = 0.21, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6728, 640, 619, 215, 742, 1025, 982, 204, 1024, 206} \[ \frac {1}{2} \sqrt {x^2+x+2} x-\frac {7}{4} \sqrt {x^2+x+2}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+x+2}}\right )}{\sqrt {3}}-\tanh ^{-1}\left (\sqrt {x^2+x+2}\right )-\frac {1}{8} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/((1 + x + x^2)*Sqrt[2 + x + x^2]),x]

[Out]

(-7*Sqrt[2 + x + x^2])/4 + (x*Sqrt[2 + x + x^2])/2 - ArcSinh[(1 + 2*x)/Sqrt[7]]/8 + ArcTan[(1 + 2*x)/(Sqrt[3]*
Sqrt[2 + x + x^2])]/Sqrt[3] - ArcTanh[Sqrt[2 + x + x^2]]

Rule 204

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

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 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 982

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

Rule 1024

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

Rule 1025

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> -Dist[(h*e - 2*g*f)/(2*f), Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/(2*f), Int[(
e + 2*f*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2
- 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && NeQ[h*e - 2*g*f, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx &=\int \left (-\frac {x}{\sqrt {2+x+x^2}}+\frac {x^2}{\sqrt {2+x+x^2}}+\frac {1+x}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}}\right ) \, dx\\ &=-\int \frac {x}{\sqrt {2+x+x^2}} \, dx+\int \frac {x^2}{\sqrt {2+x+x^2}} \, dx+\int \frac {1+x}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx\\ &=-\sqrt {2+x+x^2}+\frac {1}{2} x \sqrt {2+x+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {2+x+x^2}} \, dx+\frac {1}{2} \int \frac {-2-\frac {3 x}{2}}{\sqrt {2+x+x^2}} \, dx+\frac {1}{2} \int \frac {1}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx+\frac {1}{2} \int \frac {1+2 x}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx\\ &=-\frac {7}{4} \sqrt {2+x+x^2}+\frac {1}{2} x \sqrt {2+x+x^2}-\frac {5}{8} \int \frac {1}{\sqrt {2+x+x^2}} \, dx+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,1+2 x\right )}{2 \sqrt {7}}-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,\frac {1+2 x}{\sqrt {2+x+x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {2+x+x^2}\right )\\ &=-\frac {7}{4} \sqrt {2+x+x^2}+\frac {1}{2} x \sqrt {2+x+x^2}+\frac {1}{2} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {7}}\right )+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3} \sqrt {2+x+x^2}}\right )}{\sqrt {3}}-\tanh ^{-1}\left (\sqrt {2+x+x^2}\right )-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,1+2 x\right )}{8 \sqrt {7}}\\ &=-\frac {7}{4} \sqrt {2+x+x^2}+\frac {1}{2} x \sqrt {2+x+x^2}-\frac {1}{8} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {7}}\right )+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3} \sqrt {2+x+x^2}}\right )}{\sqrt {3}}-\tanh ^{-1}\left (\sqrt {2+x+x^2}\right )\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 134, normalized size = 1.54 \[ \frac {1}{24} \left (6 \sqrt {x^2+x+2} (2 x-7)-4 i \left (\sqrt {3}-3 i\right ) \tanh ^{-1}\left (\frac {-2 i \sqrt {3} x-i \sqrt {3}+7}{4 \sqrt {x^2+x+2}}\right )+4 i \left (\sqrt {3}+3 i\right ) \tanh ^{-1}\left (\frac {2 i \sqrt {3} x+i \sqrt {3}+7}{4 \sqrt {x^2+x+2}}\right )-3 \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/((1 + x + x^2)*Sqrt[2 + x + x^2]),x]

[Out]

(6*(-7 + 2*x)*Sqrt[2 + x + x^2] - 3*ArcSinh[(1 + 2*x)/Sqrt[7]] - (4*I)*(-3*I + Sqrt[3])*ArcTanh[(7 - I*Sqrt[3]
- (2*I)*Sqrt[3]*x)/(4*Sqrt[2 + x + x^2])] + (4*I)*(3*I + Sqrt[3])*ArcTanh[(7 + I*Sqrt[3] + (2*I)*Sqrt[3]*x)/(
4*Sqrt[2 + x + x^2])])/24

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IntegrateAlgebraic [C]  time = 1.59, size = 254, normalized size = 2.92 \[ \frac {1}{4} \sqrt {x^2+x+2} (2 x-7)+\frac {1}{8} \log \left (2 \sqrt {x^2+x+2}-2 x-1\right )+i \tan ^{-1}\left (-2 \sqrt {-\frac {1}{49}-\frac {4 i \sqrt {3}}{49}} \sqrt {x^2+x+2}+2 \sqrt {-\frac {1}{49}-\frac {4 i \sqrt {3}}{49}} x+\sqrt {-\frac {1}{49}-\frac {4 i \sqrt {3}}{49}}\right )-i \tan ^{-1}\left (-2 \sqrt {-\frac {1}{49}+\frac {4 i \sqrt {3}}{49}} \sqrt {x^2+x+2}+2 \sqrt {-\frac {1}{49}+\frac {4 i \sqrt {3}}{49}} x+\sqrt {-\frac {1}{49}+\frac {4 i \sqrt {3}}{49}}\right )-\frac {\tan ^{-1}\left (\frac {2 x^2}{\sqrt {3}}-\frac {(2 x+1) \sqrt {x^2+x+2}}{\sqrt {3}}+\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}\right )}{\sqrt {3}} \]

Warning: Unable to verify antiderivative.

[In]

IntegrateAlgebraic[(1 + x^4)/((1 + x + x^2)*Sqrt[2 + x + x^2]),x]

[Out]

((-7 + 2*x)*Sqrt[2 + x + x^2])/4 + I*ArcTan[Sqrt[-1/49 - ((4*I)/49)*Sqrt[3]] + 2*Sqrt[-1/49 - ((4*I)/49)*Sqrt[
3]]*x - 2*Sqrt[-1/49 - ((4*I)/49)*Sqrt[3]]*Sqrt[2 + x + x^2]] - I*ArcTan[Sqrt[-1/49 + ((4*I)/49)*Sqrt[3]] + 2*
Sqrt[-1/49 + ((4*I)/49)*Sqrt[3]]*x - 2*Sqrt[-1/49 + ((4*I)/49)*Sqrt[3]]*Sqrt[2 + x + x^2]] - ArcTan[2/Sqrt[3]
+ (2*x)/Sqrt[3] + (2*x^2)/Sqrt[3] - ((1 + 2*x)*Sqrt[2 + x + x^2])/Sqrt[3]]/Sqrt[3] + Log[-1 - 2*x + 2*Sqrt[2 +
x + x^2]]/8

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fricas [B]  time = 0.72, size = 147, normalized size = 1.69 \[ \frac {1}{4} \, \sqrt {x^{2} + x + 2} {\left (2 \, x - 7\right )} - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 3\right )} + \frac {2}{3} \, \sqrt {3} \sqrt {x^{2} + x + 2}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )} + \frac {2}{3} \, \sqrt {3} \sqrt {x^{2} + x + 2}\right ) + \frac {1}{2} \, \log \left (2 \, x^{2} - \sqrt {x^{2} + x + 2} {\left (2 \, x + 3\right )} + 4 \, x + 5\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - \sqrt {x^{2} + x + 2} {\left (2 \, x - 1\right )} + 3\right ) + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(x^2 + x + 2)*(2*x - 7) - 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x + 3) + 2/3*sqrt(3)*sqrt(x^2 + x + 2)) +
1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - 1) + 2/3*sqrt(3)*sqrt(x^2 + x + 2)) + 1/2*log(2*x^2 - sqrt(x^2 + x + 2
)*(2*x + 3) + 4*x + 5) - 1/2*log(2*x^2 - sqrt(x^2 + x + 2)*(2*x - 1) + 3) + 1/8*log(-2*x + 2*sqrt(x^2 + x + 2)
- 1)

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giac [B]  time = 0.68, size = 148, normalized size = 1.70 \[ \frac {1}{4} \, \sqrt {x^{2} + x + 2} {\left (2 \, x - 7\right )} - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 2 \, \sqrt {x^{2} + x + 2} + 3\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 2 \, \sqrt {x^{2} + x + 2} - 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + x + 2}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} + x + 2} + 3\right ) - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + x + 2}\right )}^{2} - x + \sqrt {x^{2} + x + 2} + 1\right ) + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(x^2 + x + 2)*(2*x - 7) - 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - 2*sqrt(x^2 + x + 2) + 3)) + 1/3*sqrt(
3)*arctan(-1/3*sqrt(3)*(2*x - 2*sqrt(x^2 + x + 2) - 1)) + 1/2*log((x - sqrt(x^2 + x + 2))^2 + 3*x - 3*sqrt(x^2
+ x + 2) + 3) - 1/2*log((x - sqrt(x^2 + x + 2))^2 - x + sqrt(x^2 + x + 2) + 1) + 1/8*log(-2*x + 2*sqrt(x^2 +
x + 2) - 1)

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maple [A]  time = 2.71, size = 63, normalized size = 0.72

method	result	size
risch	\(\frac {\left (2 x -7\right ) \sqrt {x^{2}+x +2}}{4}-\frac {\arcsinh \left (\frac {2 \sqrt {7}\, \left (\frac {1}{2}+x \right )}{7}\right )}{8}-\arctanh \left (\sqrt {x^{2}+x +2}\right )+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3 \sqrt {x^{2}+x +2}}\right ) \sqrt {3}}{3}\)	\(63\)
default	\(\frac {x \sqrt {x^{2}+x +2}}{2}-\frac {7 \sqrt {x^{2}+x +2}}{4}-\frac {\arcsinh \left (\frac {2 \sqrt {7}\, \left (\frac {1}{2}+x \right )}{7}\right )}{8}-\arctanh \left (\sqrt {x^{2}+x +2}\right )+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3 \sqrt {x^{2}+x +2}}\right ) \sqrt {3}}{3}\)	\(69\)
trager	\(\left (\frac {x}{2}-\frac {7}{4}\right ) \sqrt {x^{2}+x +2}+\frac {\ln \left (\frac {119439007666026577658978862131641122955470355864 x -1547697292817558076095613967756493895336593544 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{7}-34406958119166765379202600686515679522775898 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{9}-408428037511035132158259563133572071702902093 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{8}-15578368508457466770335635637334282174776498052 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}+92867550610519727468293762800988464367923626556 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{4}+212792550384822375335114880746026740201438299184 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{3}+273186779333881661281495026943233632964832014864 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{2}-70797359348887984486272050254218036232928952 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{9}-979164562034479840429835479647785236680079548 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{8}-4788386803733240523480198798685975575347417184 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{7}-8466366722931133143875003385747183129791862600 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{6}+10087853194776611638959898784312956615330359960 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{5}+130122211413585191690424618867539965108616302212 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{2}+196657319548613971516945068585949038000971514228 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x -338148427546681507680571202357084317563591216 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{6}+15251938730635387772703465424126946970054792756 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{5}+61460684890082525267083167131316588242892347214 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{4}+114720586388952785990125336776140922456177253704 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{3}+77433073972677421673765677678710388152097321464 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x -3249432473630521638596853264962745579360405280 x^{7}-5270431261764271464385323331931230871412749376 x^{5}+97982488562224055418949456789552431477293304320 x^{3}-8168630465209889786412283012903312887099029104 x^{6}+31249267089218217898695235629549587705149988912 x^{4}+143128220092880680254710157516723345620115609552 x^{2}-36399766591903565523888587932723614181739944 x^{9}-571567717666758708366024893119100951344256660 x^{8}-15659202392168100756065522907306187543554274712 \sqrt {x^{2}+x +2}-80541634539772649096117881567690437759482687424 x \sqrt {x^{2}+x +2}+22145439835415776771931821181847880189581078124+553367855612862471389197899373472979050440128 \sqrt {x^{2}+x +2}\, x^{7}+2940584159547113642125204402878116918693453152 \sqrt {x^{2}+x +2}\, x^{6}+6474570049250822753745477662601749042350206336 \sqrt {x^{2}+x +2}\, x^{5}-1383100890638304327135314702231314502897488800 \sqrt {x^{2}+x +2}\, x^{4}-33106204550783809819500749413074566530756750912 \sqrt {x^{2}+x +2}\, x^{3}-77646952763997029322699085858555023816147693024 \sqrt {x^{2}+x +2}\, x^{2}+36399766811592879989652699602324316481006504 \sqrt {x^{2}+x +2}\, x^{8}+22031124286478773119250323350247573703566656514 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )+34406958283933751228525684438716206247225818 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{8}+391224574382993408807496237955864608039304224 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{7}+70797359678421956184918217758619089681828792 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{8}+1336858636074842185280660610444399489138966480 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{6}+943765914223119166864375404851776970759195232 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{7}-542193641294841347379235514772028707324385008 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{5}+4269200061821671816190368251302777197688034672 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{6}-16916019799735424659186697666206970777890289580 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{4}+5896140866636764872731802925206074249112002208 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{5}-49560907705036167058886144826155835342301569264 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{3}-18379755789404327250773521187737912806531548280 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{4}-78323227078274890647304119940633201714527981192 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x^{2}-82753380069385375793447165943111704847470171136 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{3}-54753673104174503511706603527633757903688747520 \sqrt {x^{2}+x +2}\, \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )^{2} x -155975768722135705403328647704719793841721488256 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x^{2}-135163654996251933645966015919960738916686168224 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) \sqrt {x^{2}+x +2}\, x}{\left (3 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x +4 x +2\right )^{8}}\right )}{8}-\frac {\ln \left (\frac {\RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{2}+\RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x +4 x \sqrt {x^{2}+x +2}+x^{2}-5 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )+2 \sqrt {x^{2}+x +2}+x -5}{x^{2}+x +1}\right ) \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )}{2}-\ln \left (\frac {\RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x^{2}+\RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right ) x +4 x \sqrt {x^{2}+x +2}+x^{2}-5 \RootOf \left (3 \textit {\_Z}^{2}+6 \textit {\_Z} +4\right )+2 \sqrt {x^{2}+x +2}+x -5}{x^{2}+x +1}\right )\)	\(1101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(2*x-7)*(x^2+x+2)^(1/2)-1/8*arcsinh(2/7*7^(1/2)*(1/2+x))-arctanh((x^2+x+2)^(1/2))+1/3*arctan(1/3*(1+2*x)*3
^(1/2)/(x^2+x+2)^(1/2))*3^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} + 1}{\sqrt {x^{2} + x + 2} {\left (x^{2} + x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/(sqrt(x^2 + x + 2)*(x^2 + x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4+1}{\left (x^2+x+1\right )\,\sqrt {x^2+x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/((x + x^2 + 1)*(x + x^2 + 2)^(1/2)),x)

[Out]

int((x^4 + 1)/((x + x^2 + 1)*(x + x^2 + 2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} + 1}{\left (x^{2} + x + 1\right ) \sqrt {x^{2} + x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**2+x+1)/(x**2+x+2)**(1/2),x)

[Out]

Integral((x**4 + 1)/((x**2 + x + 1)*sqrt(x**2 + x + 2)), x)

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