3.12.65 \(\int \frac {\sqrt [4]{-x^3+x^4}}{-1-2 x+x^2} \, dx\) [1165]

Optimal. Leaf size=86 \[ -2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(86)=172\).
time = 0.32, antiderivative size = 365, normalized size of antiderivative = 4.24, number of steps used = 17, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2081, 919, 65, 246, 218, 212, 209, 6860, 95, 304} \begin {gather*} \frac {2 \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{10+7 \sqrt {2}} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{10-7 \sqrt {2}} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {2} \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{10+7 \sqrt {2}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {2} \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{10-7 \sqrt {2}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {2} \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) + ((10 + 7*Sqrt[2])^(1/4)*(-x^3
+ x^4)^(1/4)*ArcTan[((2 - Sqrt[2])^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[2]*(-1 + x)^(1/4)*x^(3/4)) + ((10 -
7*Sqrt[2])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[((2 + Sqrt[2])^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[2]*(-1 + x)^(1
/4)*x^(3/4)) + (2*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) - ((10 + 7*Sqrt
[2])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((2 - Sqrt[2])^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[2]*(-1 + x)^(1/4)*x
^(3/4)) - ((10 - 7*Sqrt[2])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((2 + Sqrt[2])^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(S
qrt[2]*(-1 + x)^(1/4)*x^(3/4))

Rule 65

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 95

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 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 246

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 304

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 919

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

Rule 2081

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

Rule 6860

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 {\sqrt [4]{-x^3+x^4}}{-1-2 x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-2 x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {1+x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-2 x+x^2\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {1+\sqrt {2}}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {2}+2 x\right )}+\frac {1-\sqrt {2}}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {2}+2 x\right )}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2+2 \sqrt {2}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-2-2 \sqrt {2}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \left (1-\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \left (1+\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-2-2 \sqrt {2}+2 \sqrt {2} x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{10+7 \sqrt {2}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{10-7 \sqrt {2}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {2} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{10+7 \sqrt {2}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {2} \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{10-7 \sqrt {2}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {2} \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 103, normalized size = 1.20 \begin {gather*} \frac {(-1+x)^{3/4} x^{9/4} \left (-4 \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+4 \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt [4]{x}\right )+\log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]\right )}{2 \left ((-1+x) x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 8.81, size = 3968, normalized size = 46.14

method	result	size
trager	\(\text {Expression too large to display}\)	\(3968\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)+RootOf(RootOf(2048*_Z^8-640*_
Z^4+1)^2+_Z^2)*ln((65536*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(2048*_Z^8-640*_Z^4+1)^10*x^3-13107
2*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(2048*_Z^8-640*_Z^4+1)^10*x^2+76288*RootOf(RootOf(2048*_Z^
8-640*_Z^4+1)^2+_Z^2)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^3-71680*RootOf(2048*_Z^8-640*_Z^4+1)^6*(x^4-x^3)^(1/4)*
x^2-30720*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^2-65408*(x^4-x^3)^(1/2)
*RootOf(2048*_Z^8-640*_Z^4+1)^4*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x+56448*(x^4-x^3)^(3/4)*RootOf(204
8*_Z^8-640*_Z^4+1)^4+8040*x^3*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(2048*_Z^8-640*_Z^4+1)^2-1736*
RootOf(2048*_Z^8-640*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-1800*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(204
8*_Z^8-640*_Z^4+1)^2*x^2-168*(x^4-x^3)^(1/2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x-49*(x^4-x^3)^(3/4))
/(64*x*RootOf(2048*_Z^8-640*_Z^4+1)^4-128*RootOf(2048*_Z^8-640*_Z^4+1)^4-3*x-1)/x^2)-2048/7*ln((1277952*RootOf
(2048*_Z^8-640*_Z^4+1)^9*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^
4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*x^3-2555904*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2
+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^9*x^2+42
205184*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(
2048*_Z^8-640*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^7*(x^4-x^3)^(1/2)*x-2134016*RootOf(2048*_Z^8-640*_Z^4+1)
^5*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8
-640*_Z^4+1)^2+_Z^2)*x^3+7651840*RootOf(2048*_Z^8-640*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+1891840*RootOf(2048*_Z^8-6
40*_Z^4+1)^5*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf
(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x^2-13123712*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootO
f(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^3*(x^4-x^3)^(1/2)*x
-395136*(x^4-x^3)^(3/4)*RootOf(2048*_Z^8-640*_Z^4+1)^4+539400*RootOf(2048*_Z^8-640*_Z^4+1)*RootOf(_Z^2+2240*Ro
otOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x
^3-2378264*RootOf(2048*_Z^8-640*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-341000*RootOf(2048*_Z^8-640*_Z^4+1)*RootOf(_Z^2+
2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+
_Z^2)*x^2+123137*(x^4-x^3)^(3/4))/(64*x*RootOf(2048*_Z^8-640*_Z^4+1)^4-128*RootOf(2048*_Z^8-640*_Z^4+1)^4-17*x
+41)/x^2)*RootOf(2048*_Z^8-640*_Z^4+1)^7*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2
048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)+640/7*ln((1277952*RootOf(2048*_Z^8-640*_Z^4+1)^9*Ro
otOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8
-640*_Z^4+1)^2)*x^3-2555904*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*
_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^9*x^2+42205184*RootOf(RootOf(2048*_
Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*Ro
otOf(2048*_Z^8-640*_Z^4+1)^7*(x^4-x^3)^(1/2)*x-2134016*RootOf(2048*_Z^8-640*_Z^4+1)^5*RootOf(_Z^2+2240*RootOf(
2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x^3+76
51840*RootOf(2048*_Z^8-640*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+1891840*RootOf(2048*_Z^8-640*_Z^4+1)^5*RootOf(_Z^2+22
40*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z
^2)*x^2-13123712*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-6
93*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^3*(x^4-x^3)^(1/2)*x-395136*(x^4-x^3)^(3/4)*Roo
tOf(2048*_Z^8-640*_Z^4+1)^4+539400*RootOf(2048*_Z^8-640*_Z^4+1)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^
6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x^3-2378264*RootOf(2048*_Z^8
-640*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-341000*RootOf(2048*_Z^8-640*_Z^4+1)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_
Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*x^2+123137*(x^4-x^3)^
(3/4))/(64*x*RootOf(2048*_Z^8-640*_Z^4+1)^4-128*RootOf(2048*_Z^8-640*_Z^4+1)^4-17*x+41)/x^2)*RootOf(2048*_Z^8-
640*_Z^4+1)^3*RootOf(RootOf(2048*_Z^8-640*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*
RootOf(2048*_Z^8-640*_Z^4+1)^2)+1/7*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_
Z^4+1)^2)*ln(-(1277952*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*Roo
tOf(2048*_Z^8-640*_Z^4+1)^10*x^3-2555904*RootOf...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.44, size = 496, normalized size = 5.77 \begin {gather*} \sqrt {2} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \arctan \left (\frac {{\left ({\left (3 \, \sqrt {2} x - 4 \, x\right )} \sqrt {7 \, \sqrt {2} + 10} \sqrt {-\frac {{\left (2 \, \sqrt {2} x^{2} - 3 \, x^{2}\right )} \sqrt {7 \, \sqrt {2} + 10} - \sqrt {x^{4} - x^{3}}}{x^{2}}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {7 \, \sqrt {2} + 10} {\left (3 \, \sqrt {2} - 4\right )}\right )} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \sqrt {2} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \arctan \left (\frac {{\left (3 \, \sqrt {2} x + 4 \, x\right )} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {3}{4}} \sqrt {\frac {{\left (2 \, \sqrt {2} x^{2} + 3 \, x^{2}\right )} \sqrt {-7 \, \sqrt {2} + 10} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {3}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \log \left (\frac {{\left (\sqrt {2} x - x\right )} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (\sqrt {2} x - x\right )} {\left (7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {2} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \log \left (\frac {{\left (\sqrt {2} x + x\right )} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (\sqrt {2} x + x\right )} {\left (-7 \, \sqrt {2} + 10\right )}^{\frac {1}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \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^4-x^3)^(1/4)/(x^2-2*x-1),x, algorithm="fricas")

[Out]

sqrt(2)*(7*sqrt(2) + 10)^(1/4)*arctan(1/2*((3*sqrt(2)*x - 4*x)*sqrt(7*sqrt(2) + 10)*sqrt(-((2*sqrt(2)*x^2 - 3*
x^2)*sqrt(7*sqrt(2) + 10) - sqrt(x^4 - x^3))/x^2) - (x^4 - x^3)^(1/4)*sqrt(7*sqrt(2) + 10)*(3*sqrt(2) - 4))*(7
*sqrt(2) + 10)^(1/4)/x) + sqrt(2)*(-7*sqrt(2) + 10)^(1/4)*arctan(1/2*((3*sqrt(2)*x + 4*x)*(-7*sqrt(2) + 10)^(3
/4)*sqrt(((2*sqrt(2)*x^2 + 3*x^2)*sqrt(-7*sqrt(2) + 10) + sqrt(x^4 - x^3))/x^2) - (x^4 - x^3)^(1/4)*(3*sqrt(2)
+ 4)*(-7*sqrt(2) + 10)^(3/4))/x) - 1/4*sqrt(2)*(7*sqrt(2) + 10)^(1/4)*log(((sqrt(2)*x - x)*(7*sqrt(2) + 10)^(
1/4) + (x^4 - x^3)^(1/4))/x) + 1/4*sqrt(2)*(7*sqrt(2) + 10)^(1/4)*log(-((sqrt(2)*x - x)*(7*sqrt(2) + 10)^(1/4)
- (x^4 - x^3)^(1/4))/x) - 1/4*sqrt(2)*(-7*sqrt(2) + 10)^(1/4)*log(((sqrt(2)*x + x)*(-7*sqrt(2) + 10)^(1/4) +
(x^4 - x^3)^(1/4))/x) + 1/4*sqrt(2)*(-7*sqrt(2) + 10)^(1/4)*log(-((sqrt(2)*x + x)*(-7*sqrt(2) + 10)^(1/4) - (x
^4 - x^3)^(1/4))/x) + 2*arctan((x^4 - x^3)^(1/4)/x) + log((x + (x^4 - x^3)^(1/4))/x) - log(-(x - (x^4 - x^3)^(
1/4))/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 )}}{x^{2} - 2 x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 - x^3)^(1/4)/(x^2 - 2*x - 1), 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}}{-x^2+2\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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