\(\int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx\) [1218]

Optimal result

Integrand size = 20, antiderivative size = 89 \[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\frac {1}{4} \arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {1}{4} \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 3.07 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74, number of steps used = 41, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2081, 6857, 932, 27, 6865, 1726, 1211, 1117, 1209, 1714, 1712, 212, 12, 1224, 209, 1728, 1722, 1720} \[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\frac {\sqrt {x^3+x^2+x} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}}+\frac {\sqrt {x^3+x^2+x} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{2 \sqrt {x} \sqrt {x^2+x+1}}-\frac {\sqrt {3} \sqrt {x^3+x^2+x} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}} \]

[In]

Int[Sqrt[x + x^2 + x^3]/(-1 + x^4),x]

[Out]

(Sqrt[x + x^2 + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x + x^2]])/(4*Sqrt[x]*Sqrt[1 + x + x^2]) + (Sqrt[x + x^2 + x^3]*A
rcTanh[Sqrt[x]/Sqrt[1 + x + x^2]])/(2*Sqrt[x]*Sqrt[1 + x + x^2]) - (Sqrt[3]*Sqrt[x + x^2 + x^3]*ArcTanh[(Sqrt[
3]*Sqrt[x])/Sqrt[1 + x + x^2]])/(4*Sqrt[x]*Sqrt[1 + x + x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 209

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

Rule 212

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

Rule 932

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + b*x + c*x^2]/(e*(2*m + 5))), x] - Dist[1/(e*(2*m + 5)), Int[(
(d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a
*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

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

Rule 1224

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

Rule 1712

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

Rule 1714

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

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1722

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With
[{q = Rt[c/a, 2]}, Dist[(A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x],
x] + Dist[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]
, x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2
- a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]

Rule 1726

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/e^2, Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x],
x] + Dist[1/e^2, Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a,
b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a
*e^2, 0]

Rule 1728

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]
, A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/(e*q), Int[(1 - q*x^2)/Sqrt[a + b
*x^2 + c*x^4], x], x] + Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[
a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[
c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] &&  !GtQ[b^2 - 4*a*c, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

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

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{-1+x^4} \, dx}{\sqrt {x} \sqrt {1+x+x^2}} \\ & = \frac {\sqrt {x+x^2+x^3} \int \left (-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1-x^2\right )}-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {1+x+x^2}} \\ & = -\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}} \\ & = -\frac {\sqrt {x+x^2+x^3} \int \left (\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i-x)}+\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \left (\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1-x)}+\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}} \\ & = -\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}} \\ & = \frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i-(2-2 i) x-(1-3 i) x^2}{\sqrt {x} (i+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i+(2+2 i) x+(1+3 i) x^2}{(i-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \int \frac {1+2 x^2}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {1+4 x+4 x^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}} \\ & = \frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {i-(2-2 i) x^2-(1-3 i) x^4}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {i+(2+2 i) x^2+(1+3 i) x^4}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {(1+2 x)^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1+2 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}} \\ & = -\left (-\frac {\left (\left (\frac {1}{2}-\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}\right )+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {-3-6 i x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {-3+6 i x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {3}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {\left (1+2 x^2\right )^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}} \\ & = -\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \arctan \left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}+-\frac {\left (\left (\frac {3}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {3}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {5+4 x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}} \\ & = -\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {\sqrt {x+x^2+x^3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \arctan \left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}-\frac {3 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{4 \sqrt {x} \left (1+x+x^2\right )}-\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (4 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}} \\ & = \frac {\sqrt {x+x^2+x^3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}} \\ & = \frac {\sqrt {x+x^2+x^3} \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {3} \sqrt {x+x^2+x^3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\frac {\sqrt {x} \sqrt {1+x+x^2} \left (\arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{4 \sqrt {x \left (1+x+x^2\right )}} \]

[In]

Integrate[Sqrt[x + x^2 + x^3]/(-1 + x^4),x]

[Out]

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

Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{4}-\frac {\ln \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )}{4}+\frac {\ln \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}+x}{x}\right )}{4}\)	\(83\)
pseudoelliptic	\(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{4}-\frac {\ln \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )}{4}+\frac {\ln \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}+x}{x}\right )}{4}\)	\(83\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (x -1\right )^{2}}\right )}{8}+\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}+x^{2}+x}+2 x +1}{x^{2}+1}\right )}{4}\)	\(133\)
elliptic	\(\text {Expression too large to display}\)	\(1500\)

[In]

int((x^3+x^2+x)^(1/2)/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/4*3^(1/2)*arctanh(1/3*(x*(x^2+x+1))^(1/2)/x*3^(1/2))-1/4*ln(((x*(x^2+x+1))^(1/2)-x)/x)-1/4*arctan((x*(x^2+x
+1))^(1/2)/x)+1/4*ln(((x*(x^2+x+1))^(1/2)+x)/x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\frac {1}{16} \, \sqrt {3} \log \left (\frac {x^{4} + 20 \, x^{3} - 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - \frac {1}{8} \, \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{2} + 2 \, x + 2 \, \sqrt {x^{3} + x^{2} + x} + 1}{x^{2} + 1}\right ) \]

[In]

integrate((x^3+x^2+x)^(1/2)/(x^4-1),x, algorithm="fricas")

[Out]

1/16*sqrt(3)*log((x^4 + 20*x^3 - 4*sqrt(3)*sqrt(x^3 + x^2 + x)*(x^2 + 4*x + 1) + 30*x^2 + 20*x + 1)/(x^4 - 4*x
^3 + 6*x^2 - 4*x + 1)) - 1/8*arctan(1/2*(x^2 + 1)/sqrt(x^3 + x^2 + x)) + 1/4*log((x^2 + 2*x + 2*sqrt(x^3 + x^2
+ x) + 1)/(x^2 + 1))

Sympy [F]

\[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\int \frac {\sqrt {x \left (x^{2} + x + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

integrate((x**3+x**2+x)**(1/2)/(x**4-1),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} + x^{2} + x}}{x^{4} - 1} \,d x } \]

[In]

integrate((x^3+x^2+x)^(1/2)/(x^4-1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^3 + x^2 + x)/(x^4 - 1), x)

Giac [F]

\[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} + x^{2} + x}}{x^{4} - 1} \,d x } \]

[In]

integrate((x^3+x^2+x)^(1/2)/(x^4-1),x, algorithm="giac")

[Out]

integrate(sqrt(x^3 + x^2 + x)/(x^4 - 1), x)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 565, normalized size of antiderivative = 6.35 \[ \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx=\text {Too large to display} \]

[In]

int((x + x^2 + x^3)^(1/2)/(x^4 - 1),x)

[Out]

(((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^
(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(- 3^(1/2)/2 - 1i/2, asin((x/((3^(1/
2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*((3^(1/2)*1i)/2 -
1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)
*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*elliptic
Pi(3^(1/2)/2 + 1i/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/
(2*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)
*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((
3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(1/2 - (3^(1/2)*1i)/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2
)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)
) - (3*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 -
1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 - 1/2, asin((x
/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*((3^(1/2)*
1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))