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

Optimal. Leaf size=158 \[ -9 \text {RootSum}\left [2 \text {$\#$1}^8-6 \text {$\#$1}^4+3\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]+\frac {1}{4} \sqrt [4]{x^4-x^3} (4 x+11)-\frac {177}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {177}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]

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Rubi [B]  time = 1.02, antiderivative size = 678, normalized size of antiderivative = 4.29, number of steps used = 34, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2056, 6728, 50, 63, 240, 212, 206, 203, 101, 157, 93, 298} \begin {gather*} -\sqrt [4]{x^4-x^3} (1-x)+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{4} \sqrt [4]{x^4-x^3}+\frac {3 \left (15+7 \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {3 \left (15-7 \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {3 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {3 \left (15+7 \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {3 \left (15-7 \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {3 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*(-x^3 + x^4)^(1/4))/4 + (3*(1 - Sqrt[3])*(-x^3 + x^4)^(1/4))/2 + (3*(1 + Sqrt[3])*(-x^3 + x^4)^(1/4))/2 - (
1 - x)*(-x^3 + x^4)^(1/4) - (3*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(8*(-1 + x)^(1/4)*x^(3/4)) +
(3*(15 - 7*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + (3*(15 +
7*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + (3*2^(3/4)*(123 + 7
1*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[((3/(3 + Sqrt[3]))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*
x^(3/4)) + (3*2^(3/4)*(123 - 71*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[(((3 + Sqrt[3])/2)^(1/4)*x^(1/4))/(-1
+ x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) - (3*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(8*(-1 + x)^(1
/4)*x^(3/4)) + (3*(15 - 7*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/
4)) + (3*(15 + 7*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) - (3*
2^(3/4)*(123 + 71*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((3/(3 + Sqrt[3]))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])
/((-1 + x)^(1/4)*x^(3/4)) - (3*2^(3/4)*(123 - 71*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(((3 + Sqrt[3])/2)^
(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-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 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 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 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 (2-x+2 x^2\right ) \sqrt [4]{-x^3+x^4}}{-2-2 x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (2-x+2 x^2\right )}{-2-2 x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (2 \sqrt [4]{-1+x} x^{3/4}+\frac {3 \sqrt [4]{-1+x} x^{3/4} (2+x)}{-2-2 x+x^2}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \sqrt [4]{-1+x} x^{3/4} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4} (2+x)}{-2-2 x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\left ((1-x) \sqrt [4]{-x^3+x^4}\right )+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-2-2 \sqrt {3}+2 x}+\frac {\left (1-\sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-2+2 \sqrt {3}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^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}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2+2 \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-2-2 \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2} \left (1-\sqrt {3}\right )+\frac {1}{2} \left (-3+4 \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{2} \left (1+\sqrt {3}\right )+\frac {1}{2} \left (-3-4 \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+4 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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 )}{8 \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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (12 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {3}-2 \sqrt {3} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (12 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2-2 \sqrt {3}+2 \sqrt {3} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \sqrt {3}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+4 \sqrt {3}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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 )}{8 \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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {3}}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (3-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {3}}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {3}}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {3}}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \sqrt {3}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+4 \sqrt {3}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\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 )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \sqrt {3}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \left (1-\sqrt {3}\right ) \left (-3+4 \sqrt {3}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+4 \sqrt {3}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (3 \left (1+\sqrt {3}\right ) \left (3+4 \sqrt {3}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1-\sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {3}{2} \left (1+\sqrt {3}\right ) \sqrt [4]{-x^3+x^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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15-7 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15+7 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\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 )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15-7 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {3 \left (15+7 \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123+71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{3+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3\ 2^{3/4} \sqrt [4]{123-71 \sqrt {3}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 158, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (15 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+2 (x-1) \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )+15 \left (3 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+\left (\sqrt {3}-2\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {\left (-3+\sqrt {3}\right ) (x-1)}{3 x}\right )-\left (2+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (3+\sqrt {3}\right ) (x-1)}{3 x}\right )\right )\right )}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-1 + x)*x^3)^(1/4)*(15*x^(1/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, 1 - x] + 2*(-1 + x)*x^(1/4)*Hypergeometr
ic2F1[-3/4, 5/4, 9/4, 1 - x] + 15*(3*x^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, 1 - x] + (-2 + Sqrt[3])*Hypergeo
metric2F1[1/4, 1, 5/4, -1/3*((-3 + Sqrt[3])*(-1 + x))/x] - (2 + Sqrt[3])*Hypergeometric2F1[1/4, 1, 5/4, ((3 +
Sqrt[3])*(-1 + x))/(3*x)])))/(5*x)

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IntegrateAlgebraic [A]  time = 0.00, size = 158, normalized size = 1.00 \begin {gather*} \frac {1}{4} (11+4 x) \sqrt [4]{-x^3+x^4}-\frac {177}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {177}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-9 \text {RootSum}\left [3-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((11 + 4*x)*(-x^3 + x^4)^(1/4))/4 - (177*ArcTan[x/(-x^3 + x^4)^(1/4)])/8 + (177*ArcTanh[x/(-x^3 + x^4)^(1/4)])
/8 - 9*RootSum[3 - 6*#1^4 + 2*#1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3
+ x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]

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fricas [B]  time = 0.53, size = 590, normalized size = 3.73 \begin {gather*} -6 \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {71 \, \sqrt {3} + 123} {\left (19 \, \sqrt {3} - 33\right )} - {\left (19 \, \sqrt {3} x - 33 \, x\right )} \sqrt {71 \, \sqrt {3} + 123} \sqrt {-\frac {\sqrt {2} {\left (4 \, \sqrt {3} x^{2} - 7 \, x^{2}\right )} \sqrt {71 \, \sqrt {3} + 123} - 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}}\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}}}{12 \, x}\right ) + \frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \log \left (\frac {3 \, {\left (\sqrt {2} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} \log \left (-\frac {3 \, {\left (\sqrt {2} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {2} \sqrt {71 \, \sqrt {3} + 123}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + 2 \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \arctan \left (-\frac {3 \, \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (19 \, \sqrt {3} + 33\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {3}{4}} - {\left (19 \, \sqrt {3} x + 33 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {3}{4}} \sqrt {\frac {{\left (4 \, \sqrt {3} x^{2} + 7 \, x^{2}\right )} \sqrt {-11502 \, \sqrt {3} + 19926} + 18 \, \sqrt {x^{4} - x^{3}}}{x^{2}}}}{972 \, x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} + 6 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} {\left (-11502 \, \sqrt {3} + 19926\right )}^{\frac {1}{4}} - 6 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 11\right )} + \frac {177}{8} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {177}{16} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {177}{16} \, \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((2*x^2-x+2)*(x^4-x^3)^(1/4)/(x^2-2*x-2),x, algorithm="fricas")

[Out]

-6*sqrt(2)*sqrt(sqrt(2)*sqrt(71*sqrt(3) + 123))*arctan(-1/12*sqrt(2)*(sqrt(2)*(x^4 - x^3)^(1/4)*sqrt(71*sqrt(3
) + 123)*(19*sqrt(3) - 33) - (19*sqrt(3)*x - 33*x)*sqrt(71*sqrt(3) + 123)*sqrt(-(sqrt(2)*(4*sqrt(3)*x^2 - 7*x^
2)*sqrt(71*sqrt(3) + 123) - 2*sqrt(x^4 - x^3))/x^2))*sqrt(sqrt(2)*sqrt(71*sqrt(3) + 123))/x) + 3/2*sqrt(2)*sqr
t(sqrt(2)*sqrt(71*sqrt(3) + 123))*log(3*(sqrt(2)*(sqrt(3)*x - 2*x)*sqrt(sqrt(2)*sqrt(71*sqrt(3) + 123)) + 2*(x
^4 - x^3)^(1/4))/x) - 3/2*sqrt(2)*sqrt(sqrt(2)*sqrt(71*sqrt(3) + 123))*log(-3*(sqrt(2)*(sqrt(3)*x - 2*x)*sqrt(
sqrt(2)*sqrt(71*sqrt(3) + 123)) - 2*(x^4 - x^3)^(1/4))/x) + 2*sqrt(2)*(-11502*sqrt(3) + 19926)^(1/4)*arctan(-1
/972*(3*sqrt(2)*(x^4 - x^3)^(1/4)*(19*sqrt(3) + 33)*(-11502*sqrt(3) + 19926)^(3/4) - (19*sqrt(3)*x + 33*x)*(-1
1502*sqrt(3) + 19926)^(3/4)*sqrt(((4*sqrt(3)*x^2 + 7*x^2)*sqrt(-11502*sqrt(3) + 19926) + 18*sqrt(x^4 - x^3))/x
^2))/x) - 1/2*sqrt(2)*(-11502*sqrt(3) + 19926)^(1/4)*log((sqrt(2)*(sqrt(3)*x + 2*x)*(-11502*sqrt(3) + 19926)^(
1/4) + 6*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(2)*(-11502*sqrt(3) + 19926)^(1/4)*log(-(sqrt(2)*(sqrt(3)*x + 2*x)*(-
11502*sqrt(3) + 19926)^(1/4) - 6*(x^4 - x^3)^(1/4))/x) + 1/4*(x^4 - x^3)^(1/4)*(4*x + 11) + 177/8*arctan((x^4
- x^3)^(1/4)/x) + 177/16*log((x + (x^4 - x^3)^(1/4))/x) - 177/16*log(-(x - (x^4 - x^3)^(1/4))/x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [B]  time = 31.11, size = 3305, normalized size = 20.92

method	result	size
trager	\(\text {Expression too large to display}\)	\(3305\)
risch	\(\text {Expression too large to display}\)	\(6772\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11/4+x)*(x^4-x^3)^(1/4)-177/16*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
)+3/4*RootOf(_Z^8-31488*_Z^4+98304)*ln((11*RootOf(_Z^8-31488*_Z^4+98304)^11*x^3-22*RootOf(_Z^8-31488*_Z^4+9830
4)^11*x^2+1162240*RootOf(_Z^8-31488*_Z^4+98304)^7*x^3+274688*RootOf(_Z^8-31488*_Z^4+98304)^7*x^2-1308672*RootO
f(_Z^8-31488*_Z^4+98304)^6*(x^4-x^3)^(1/4)*x^2-13959168*(x^4-x^3)^(1/2)*RootOf(_Z^8-31488*_Z^4+98304)^5*x+2442
85440*(x^4-x^3)^(3/4)*RootOf(_Z^8-31488*_Z^4+98304)^4-50510168064*RootOf(_Z^8-31488*_Z^4+98304)^3*x^3+15003942
912*RootOf(_Z^8-31488*_Z^4+98304)^3*x^2+76105383936*RootOf(_Z^8-31488*_Z^4+98304)^2*(x^4-x^3)^(1/4)*x^2-815215
41120*(x^4-x^3)^(1/2)*RootOf(_Z^8-31488*_Z^4+98304)*x+86658514944*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-31488*_Z^4+9
8304)^4-2*RootOf(_Z^8-31488*_Z^4+98304)^4-6656*x-4864)/x^2)-3/4*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*l
n((11*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^10*x^3-22*RootOf(RootOf(_Z^8-
31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^10*x^2+1162240*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+
_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^6*x^3+274688*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-3148
8*_Z^4+98304)^6*x^2+1308672*RootOf(_Z^8-31488*_Z^4+98304)^6*(x^4-x^3)^(1/4)*x^2+13959168*(x^4-x^3)^(1/2)*RootO
f(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^4*x+244285440*(x^4-x^3)^(3/4)*RootOf(_Z^
8-31488*_Z^4+98304)^4-50510168064*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^2
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1488)^2*RootOf(_Z^8-31488*_Z^4+98304)^4*(x^4-x^3)^(1/4)*x^2-21361993973760*(x^4-x^3)^(1/2)*RootOf(_Z^8-31488*_
Z^4+98304)^3*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^4+_Z^4-31488)*x
-31268536320*(x^4-x^3)^(3/4)*RootOf(_Z^8-31488*_Z^4+98304)^4-4466933760000*RootOf(RootOf(_Z^8-31488*_Z^4+98304
)^4+_Z^4-31488)^2*(x^4-x^3)^(1/4)*x^2+995675961556992*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-31488*_Z^4+98304)^4-2*Ro
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*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^4+_Z^4-31488)^3*RootOf(_Z^8
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Z^2)*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^4+_Z^4-31488)^3*RootOf(_Z^8-31488*_Z^4+98304)^7+678473728*(x^4-x^3)^
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488)^3*RootOf(_Z^8-31488*_Z^4+98304)^8*x^3-22*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^4+_Z^4-31488)^3*RootOf(_Z^8
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^8-31488*_Z^4+98304)^4-24832*x+67840)/x^2)-59/524288*ln(-(2*(x^4-x^3)^(1/2)*RootOf(_Z^8-31488*_Z^4+98304)^7*Ro
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2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^3*x^2-196608*(x^4-x^3)^(3/4)+196608*x^2*(x^4-x^3)^(1/4))/x^2)*RootOf(_Z^
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8-31488*_Z^4+98304)^7*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^2)*x-2*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+
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(_Z^8-31488*_Z^4+98304)^2+_Z^2)*RootOf(_Z^8-31488*_Z^4+98304)^3*x^2-196608*(x^4-x^3)^(3/4)+196608*x^2*(x^4-x^3
)^(1/4))/x^2)*RootOf(_Z^8-31488*_Z^4+98304)^3*RootOf(RootOf(_Z^8-31488*_Z^4+98304)^2+_Z^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 (2 \, x^{2} - x + 2\right )}}{x^{2} - 2 \, x - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (x^4-x^3\right )}^{1/4}\,\left (2\,x^2-x+2\right )}{-x^2+2\,x+2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(-((x^4 - x^3)^(1/4)*(2*x^2 - x + 2))/(2*x - x^2 + 2), 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 (2 x^{2} - x + 2\right )}{x^{2} - 2 x - 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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