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

Optimal. Leaf size=230 \[ \frac {1}{16} \sqrt [4]{x^4-x^3} (4 x-5)-\frac {57}{32} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {57}{32} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4-x^3}}{\sqrt {x^4-x^3}-x^2}\right )}{12 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^4-x^3}}{\sqrt {2}}}{x \sqrt [4]{x^4-x^3}}\right )}{12 \sqrt {2}} \]

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Rubi [B]  time = 1.05, antiderivative size = 480, normalized size of antiderivative = 2.09, number of steps used = 40, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {2056, 6728, 50, 63, 240, 212, 206, 203, 101, 157, 93, 297, 1162, 617, 204, 1165, 628, 298} \begin {gather*} -\frac {1}{4} \sqrt [4]{x^4-x^3} (1-x)-\frac {1}{16} \sqrt [4]{x^4-x^3}+\frac {5 \sqrt [4]{x^4-x^3} \log \left (\frac {\sqrt {x}}{\sqrt {x-1}}-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{24 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}-\frac {5 \sqrt [4]{x^4-x^3} \log \left (\frac {\sqrt {x}}{\sqrt {x-1}}+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{24 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}-\frac {5 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{12 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {5 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{x-1}}+1\right )}{12 \sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {57 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{x-1} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {57 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{x-1} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(-x^3 + x^4)^(1/4) - ((1 - x)*(-x^3 + x^4)^(1/4))/4 - (5*(-x^3 + x^4)^(1/4)*ArcTan[1 - (Sqrt[2]*x^(1/4))
/(-1 + x)^(1/4)])/(12*Sqrt[2]*(-1 + x)^(1/4)*x^(3/4)) + (5*(-x^3 + x^4)^(1/4)*ArcTan[1 + (Sqrt[2]*x^(1/4))/(-1
+ x)^(1/4)])/(12*Sqrt[2]*(-1 + x)^(1/4)*x^(3/4)) + (57*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(32
*(-1 + x)^(1/4)*x^(3/4)) + (4*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)
^(1/4)*x^(3/4)) + (57*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(32*(-1 + x)^(1/4)*x^(3/4)) - (4*2^(
1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (5*(-x^3 + x^4
)^(1/4)*Log[1 - (Sqrt[2]*x^(1/4))/(-1 + x)^(1/4) + Sqrt[x]/Sqrt[-1 + x]])/(24*Sqrt[2]*(-1 + x)^(1/4)*x^(3/4))
- (5*(-x^3 + x^4)^(1/4)*Log[1 + (Sqrt[2]*x^(1/4))/(-1 + x)^(1/4) + Sqrt[x]/Sqrt[-1 + x]])/(24*Sqrt[2]*(-1 + x)
^(1/4)*x^(3/4))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 203

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

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

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

Rule 1165

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

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 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 {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (1+x^2\right )}{-1+x+2 x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {1}{2} \sqrt [4]{-1+x} x^{3/4}+\frac {(3-x) \sqrt [4]{-1+x} x^{3/4}}{2 \left (-1+x+2 x^2\right )}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \sqrt [4]{-1+x} x^{3/4} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {(3-x) \sqrt [4]{-1+x} x^{3/4}}{-1+x+2 x^2} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {10 \sqrt [4]{-1+x} x^{3/4}}{3 (-2+4 x)}-\frac {16 \sqrt [4]{-1+x} x^{3/4}}{3 (4+4 x)}\right ) \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2+4 x} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (8 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{4+4 x} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2}-x}{(-1+x)^{3/4} \sqrt [4]{x} (-2+4 x)} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {-3+5 x}{(-1+x)^{3/4} \sqrt [4]{x} (4+4 x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{48 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (-2+4 x)} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (16 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (4+4 x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{12 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (10 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (64 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{4-8 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{-2-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{6 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (10 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}+\frac {57 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (5 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{16} \sqrt [4]{-x^3+x^4}-\frac {1}{4} (1-x) \sqrt [4]{-x^3+x^4}-\frac {5 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{12 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {57 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}+\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1-\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {5 \sqrt [4]{-x^3+x^4} \log \left (1+\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}+\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{24 \sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.18, size = 126, normalized size = 0.55 \begin {gather*} \frac {\sqrt [4]{(x-1) x^3} \left (-30 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+12 (x-1) \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )+5 \left (27 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )-5 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1}{x}-1\right )-16 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{2 x}\right )\right )\right )}{30 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-1 + x)*x^3)^(1/4)*(-30*x^(1/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, 1 - x] + 12*(-1 + x)*x^(1/4)*Hypergeometr
ic2F1[-3/4, 5/4, 9/4, 1 - x] + 5*(27*x^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, 1 - x] - 5*Hypergeometric2F1[1/4
, 1, 5/4, -1 + x^(-1)] - 16*Hypergeometric2F1[1/4, 1, 5/4, (-1 + x)/(2*x)])))/(30*x)

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IntegrateAlgebraic [A]  time = 0.69, size = 230, normalized size = 1.00 \begin {gather*} \frac {1}{16} (-5+4 x) \sqrt [4]{-x^3+x^4}-\frac {57}{32} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^3+x^4}}{-x^2+\sqrt {-x^3+x^4}}\right )}{12 \sqrt {2}}+\frac {57}{32} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {5 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x^3+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x^3+x^4}}\right )}{12 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5 + 4*x)*(-x^3 + x^4)^(1/4))/16 - (57*ArcTan[x/(-x^3 + x^4)^(1/4)])/32 + (4*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^
3 + x^4)^(1/4)])/3 + (5*ArcTan[(Sqrt[2]*x*(-x^3 + x^4)^(1/4))/(-x^2 + Sqrt[-x^3 + x^4])])/(12*Sqrt[2]) + (57*A
rcTanh[x/(-x^3 + x^4)^(1/4)])/32 - (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - (5*ArcTanh[(x^2/Sqr
t[2] + Sqrt[-x^3 + x^4]/Sqrt[2])/(x*(-x^3 + x^4)^(1/4))])/(12*Sqrt[2])

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fricas [B]  time = 0.58, size = 423, normalized size = 1.84 \begin {gather*} \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}} - x - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}} + x - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{16} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 5\right )} + \frac {8}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{32} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{64} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {57}{64} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/12*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^4 - x^3)^(1/4)*x + sqrt(x^4 - x^3))/x^2) - x - sqrt(2)*(
x^4 - x^3)^(1/4))/x) + 5/12*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^4 - x^3)^(1/4)*x + sqrt(x^4 - x^3
))/x^2) + x - sqrt(2)*(x^4 - x^3)^(1/4))/x) - 5/48*sqrt(2)*log(4*(x^2 + sqrt(2)*(x^4 - x^3)^(1/4)*x + sqrt(x^4
- x^3))/x^2) + 5/48*sqrt(2)*log(4*(x^2 - sqrt(2)*(x^4 - x^3)^(1/4)*x + sqrt(x^4 - x^3))/x^2) + 1/16*(x^4 - x^
3)^(1/4)*(4*x - 5) + 8/3*2^(1/4)*arctan(1/2*(2^(3/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 - x^3))/x^2) - 2^(3/4)*(x^
4 - x^3)^(1/4))/x) - 2/3*2^(1/4)*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 -
x^3)^(1/4))/x) + 57/32*arctan((x^4 - x^3)^(1/4)/x) + 57/64*log((x + (x^4 - x^3)^(1/4))/x) - 57/64*log(-(x - (
x^4 - x^3)^(1/4))/x)

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giac [A]  time = 0.48, size = 244, normalized size = 1.06 \begin {gather*} -\frac {1}{16} \, {\left (5 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} + \frac {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {5}{24} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{24} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {5}{48} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {57}{32} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {57}{64} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {57}{64} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(5*(-1/x + 1)^(5/4) - (-1/x + 1)^(1/4))*x^2 + 1/3*8^(3/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) + 5/24*sq
rt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-1/x + 1)^(1/4))) + 5/24*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-1/x
+ 1)^(1/4))) + 5/48*sqrt(2)*log(sqrt(2)*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 5/48*sqrt(2)*log(-sqrt(2)*(-
1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 2/3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) - 2/3*2^(1/4)*log(abs(-2^(1
/4) + (-1/x + 1)^(1/4))) - 57/32*arctan((-1/x + 1)^(1/4)) - 57/64*log((-1/x + 1)^(1/4) + 1) + 57/64*log(abs((-
1/x + 1)^(1/4) - 1))

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maple [C]  time = 5.30, size = 851, normalized size = 3.70

method	result	size
trager	\(\left (-\frac {5}{16}+\frac {x}{4}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{3}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )}{3}+\frac {57 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{128}-\frac {5 \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (-1+2 x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{48}+\frac {5 \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (-1+2 x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{48}-\frac {5 \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{4}-x^{3}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+2 x \right )}\right )}{24}-\frac {57 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{64}\)	\(851\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5/16+1/4*x)*(x^4-x^3)^(1/4)-2/3*RootOf(_Z^4-2)*ln((3*RootOf(_Z^4-2)^3*x^3+4*(x^4-x^3)^(1/4)*RootOf(_Z^4-2)^2
*x^2-RootOf(_Z^4-2)^3*x^2+4*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)*x+4*(x^4-x^3)^(3/4))/x^2/(1+x))-2/3*RootOf(_Z^2+Roo
tOf(_Z^4-2)^2)*ln((-3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^3+RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(
_Z^4-2)^2*x^2-4*(x^4-x^3)^(1/4)*RootOf(_Z^4-2)^2*x^2+4*(x^4-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x+4*(x^4-
x^3)^(3/4))/x^2/(1+x))+57/128*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln((2*(x^4-x^3)^(1/2)*RootOf(_Z^4
-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x-2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^3+RootOf(_Z^2+RootOf(
_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^2+4*(x^4-x^3)^(3/4)-4*x^2*(x^4-x^3)^(1/4))/x^2)-5/48*ln((4*RootOf(_Z^4-2)^2*(x^4
-x^3)^(1/2)*x+2*RootOf(_Z^4-2)^2*x^3-RootOf(_Z^4-2)^2*x^2+4*(x^4-x^3)^(3/4)+4*x^2*(x^4-x^3)^(1/4))/x^2/(-1+2*x
))*RootOf(_Z^4-2)^2+5/48*ln((4*RootOf(_Z^4-2)^2*(x^4-x^3)^(1/2)*x+2*RootOf(_Z^4-2)^2*x^3-RootOf(_Z^4-2)^2*x^2+
4*(x^4-x^3)^(3/4)+4*x^2*(x^4-x^3)^(1/4))/x^2/(-1+2*x))*RootOf(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)-5/24*RootO
f(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln((2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*(x^4-x^3)^(1/4)*RootOf(_Z^4-2)^3*x
^2+2*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x+2*RootOf(_Z^4-2)^2*(x^4-x^3)^(1/2)*x+RootO
f(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)*x^2-RootOf(_Z^4-2)^2*x^2+4*(x^4-x^3)^(3/4))/x^2/(-1+2*x))-57/64*ln((2*
(x^4-x^3)^(3/4)-2*(x^4-x^3)^(1/2)*x+2*x^2*(x^4-x^3)^(1/4)-2*x^3+x^2)/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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