∫ 4 √ _____ −x3+x4

3.12.64 $\int \frac {\sqrt [4]{-x^3+x^4}}{-1-2 x+x^2} \, dx$

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

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Rubi [B] time = 0.70, 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 = {2056, 905, 63, 240, 212, 206, 203, 6728, 93, 298} \begin {gather*} \frac {2 \sqrt [4]{x^4-x^3} \tan ^{-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} \tan ^{-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} \tan ^{-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 {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*}

Warning: Unable to verify antiderivative.

[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 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 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 905

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 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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.11, size = 105, normalized size = 1.22 \begin {gather*} \frac {\sqrt [4]{(x-1) x^3} \left (4 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+\left (\sqrt {2}-2\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {\left (-2+\sqrt {2}\right ) (x-1)}{2 x}\right )-\left (2+\sqrt {2}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (2+\sqrt {2}\right ) (x-1)}{2 x}\right )\right )}{x} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(((-1 + x)*x^3)^(1/4)*(4*x^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, 1 - x] + (-2 + Sqrt[2])*Hypergeometric2F1[1/

4, 1, 5/4, -1/2*((-2 + Sqrt[2])*(-1 + x))/x] - (2 + Sqrt[2])*Hypergeometric2F1[1/4, 1, 5/4, ((2 + Sqrt[2])*(-1

+ x))/(2*x)]))/x

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IntegrateAlgebraic [A] time = 0.28, size = 86, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

og[(-x^3 + x^4)^(1/4) - x*#1])/#1^3 & ]/2

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fricas [B] time = 0.97, 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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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((x^4-x^3)^(1/4)/(x^2-2*x-1),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 = 13.65, size = 3910, normalized size = 45.47


method	result	size

trager	$\text {Expression too large to display}$	$3910$

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(_Z^2+1)*ln((2*(x^4-x^3

)^(1/2)*RootOf(_Z^2+1)*x-2*RootOf(_Z^2+1)*x^3+2*(x^4-x^3)^(3/4)-2*x^2*(x^4-x^3)^(1/4)+RootOf(_Z^2+1)*x^2)/x^2)

-2048/343*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(_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^3+2555904*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^8-640*_Z^4+1)^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+303

9232*(x^4-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+1)^7*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf

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x-1335296*x^3*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)^5*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)+294400*

x^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)^5*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-3512320*(x^4-x^3)

^(1/4)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^2-17024*(x^4-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+1)^3*RootOf(_Z^2-224

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)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-600*x^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)*RootOf(_Z^2+2

240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-85064*(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-

640*_Z^4+1)^2*x^2-2401*(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)*RootOf(2048*_Z^8-640*_Z^4+1)^7*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_

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*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)*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^3+255

5904*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(_Z^2+2240*Root

Of(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+3039232*(x^4

-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+1)^7*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^

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*x^3*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)^5*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)+294400*x^2*RootO

f(_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)^5*

RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-3512320*(x^4-x^3)^(1/4)*Ro

otOf(2048*_Z^8-640*_Z^4+1)^6*x^2-17024*(x^4-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+1)^3*RootOf(_Z^2-2240*RootOf(

2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-69

3*RootOf(2048*_Z^8-640*_Z^4+1)^2)*x+2765952*(x^4-x^3)^(3/4)*RootOf(2048*_Z^8-640*_Z^4+1)^4+2680*x^3*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)*RootOf(

_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-600*x^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)*RootOf(2048*_Z^8-640*_Z^4+1)*RootOf(_Z^2+2240*RootO

f(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)-85064*(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-640*_Z^4+

1)^2*x^2-2401*(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)*RootOf(2048*_Z^8-640*_Z^4+1)^3*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^8-640*_

Z^4+1)^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)*RootOf(2048*_Z^8-640*_Z^4+1)^10*x^3+2555904*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)^10*

x^2+2134016*x^3*RootOf(_Z^2+2240*RootOf(2048*_Z^8-640*_Z^4+1)^6-693*RootOf(2048*_Z^8-640*_Z^4+1)^2)*RootOf(204

8*_Z^8-640*_Z^4+1)^6+7651840*(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^2-1891840*RootOf(_Z^2+2240*RootO

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f(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^8-6

40*_Z^4+1)^2)*RootOf(2048*_Z^8-640*_Z^4+1)^10*x^3-2555904*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)^10*x^2-2134016*x^3*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)^6+1891840*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)^6*x^2-7651840*

(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^2+65408*(x^4-x^3)^(1/2)*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)^4*x+395136*(x^4-x^3)^(3/4)*RootOf(

2048*_Z^8-640*_Z^4+1)^4+539400*x^3*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)^2-341000*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)^2*x^2+2378264*(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-640*_Z^4+1)^2

*x^2-20608*(x^4-x^3)^(1/2)*RootOf(_Z^2-2240*RootOf(2048*_Z^8-640*_Z^4+1)^6+693*RootOf(2048*_Z^8-640*_Z^4+1)^2)

*x-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)*ln((65536*RootOf(2048*_Z^8-640*_Z^4+1)^11*x^3-131072*x^2*RootOf(2048*_Z^8-640*

_Z^4+1)^11+76288*RootOf(2048*_Z^8-640*_Z^4+1)^7*x^3+71680*(x^4-x^3)^(1/4)*RootOf(2048*_Z^8-640*_Z^4+1)^6*x^2-3

0720*RootOf(2048*_Z^8-640*_Z^4+1)^7*x^2+65408*(x^4-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+1)^5*x+56448*(x^4-x^3)

^(3/4)*RootOf(2048*_Z^8-640*_Z^4+1)^4+8040*x^3*RootOf(2048*_Z^8-640*_Z^4+1)^3+1736*(x^4-x^3)^(1/4)*RootOf(2048

*_Z^8-640*_Z^4+1)^2*x^2-1800*RootOf(2048*_Z^8-640*_Z^4+1)^3*x^2+168*(x^4-x^3)^(1/2)*RootOf(2048*_Z^8-640*_Z^4+

1)*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)

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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}}}{x^{2} - 2 \, x - 1}\,{d x} \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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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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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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3.12.64 \(\int \frac {\sqrt [4]{-x^3+x^4}}{-1-2 x+x^2} \, dx\)