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

Optimal. Leaf size=89 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{4 \sqrt {3}} \]

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Rubi [C]  time = 2.39, antiderivative size = 541, normalized size of antiderivative = 6.08, number of steps used = 59, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2056, 6725, 957, 716, 1103, 934, 169, 538, 537, 1139, 1195} \begin {gather*} \frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[(
(1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (Sqrt[x]*Sq
rt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 - I*Sqrt[3])/2, ArcSin[((1 - I*Sqr
t[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (Sqrt[x]*Sqrt[1 + (2*
x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-1 + I*Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqr
t[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) + (Sqrt[x]*Sqrt[1 + (2*x)/(1 - I
*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(I + Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I
+ Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3])

Rule 169

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

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

Rule 716

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

Rule 934

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

Rule 957

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Int
[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b
, c, d, e, f, g}, 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[
n + 1/2]

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1139

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

Rule 1195

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), 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 2056

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 6725

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]

Rubi steps

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

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Mathematica [C]  time = 0.62, size = 153, normalized size = 1.72 \begin {gather*} \frac {\sqrt [3]{-1} \sqrt {\sqrt [3]{-1} x \left (x+\sqrt [3]{-1}\right )} \sqrt {\sqrt [3]{-1} x+1} \left (\Pi \left (-\sqrt [3]{-1};\sin ^{-1}\left (\sqrt {(-1)^{2/3} x}\right )|(-1)^{2/3}\right )+\Pi \left (\sqrt [3]{-1};\sin ^{-1}\left (\sqrt {(-1)^{2/3} x}\right )|(-1)^{2/3}\right )-\Pi \left (-(-1)^{5/6};\sin ^{-1}\left (\sqrt {(-1)^{2/3} x}\right )|(-1)^{2/3}\right )-\Pi \left ((-1)^{5/6};\sin ^{-1}\left (\sqrt {(-1)^{2/3} x}\right )|(-1)^{2/3}\right )\right )}{2 \sqrt {x \left (x^2+x+1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/3)*Sqrt[(-1)^(1/3)*x*((-1)^(1/3) + x)]*Sqrt[1 + (-1)^(1/3)*x]*(EllipticPi[-(-1)^(1/3), ArcSin[Sqrt[(-
1)^(2/3)*x]], (-1)^(2/3)] + EllipticPi[(-1)^(1/3), ArcSin[Sqrt[(-1)^(2/3)*x]], (-1)^(2/3)] - EllipticPi[-(-1)^
(5/6), ArcSin[Sqrt[(-1)^(2/3)*x]], (-1)^(2/3)] - EllipticPi[(-1)^(5/6), ArcSin[Sqrt[(-1)^(2/3)*x]], (-1)^(2/3)
]))/(2*Sqrt[x*(1 + x + x^2)])

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IntegrateAlgebraic [A]  time = 0.15, size = 89, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.29, size = 120, normalized size = 1.35 \begin {gather*} \frac {1}{48} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} + x^{2} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.81, size = 133, normalized size = 1.49

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {x^{3}+x^{2}+x}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{\left (-1+x \right )^{2}}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{8}+\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}+x^{2}+x}+2 x +1}{x^{2}+1}\right )}{4}\)	\(133\)
default	\(-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(960\)
elliptic	\(-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(1500\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*RootOf(_Z^2-3)*ln((RootOf(_Z^2-3)*x^2+4*RootOf(_Z^2-3)*x+6*(x^3+x^2+x)^(1/2)+RootOf(_Z^2-3))/(-1+x)^2)+1
/8*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*x^2+2*(x^3+x^2+x)^(1/2)-RootOf(_Z^2+1))/(1+x)^2)+1/4*ln((x^2+2*(x^3+x^2+
x)^(1/2)+2*x+1)/(x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} + x^{2} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 565, normalized size = 6.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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