3.28.62 \(\int \frac {(-1+x+2 x^2) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx\) [2762]

Optimal. Leaf size=261 \[ \frac {1}{4} (11+4 x) \sqrt [4]{-x^3+x^4}-\frac {49}{8} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {2 \left (11+5 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {2 \left (-11+5 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {49}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {2 \left (11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {2 \left (-11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right ) \]

[Out]

1/4*(11+4*x)*(x^4-x^3)^(1/4)-49/8*arctan(x/(x^4-x^3)^(1/4))+(22+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1
/2)*x/(x^4-x^3)^(1/4))-(-22+10*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))+49/8*arctanh(x
/(x^4-x^3)^(1/4))-(22+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))+(-22+10*5^(1/2))^(
1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(672\) vs. \(2(261)=522\).
time = 0.78, antiderivative size = 672, normalized size of antiderivative = 2.57, number of steps used = 34, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2081, 6860, 52, 65, 246, 218, 212, 209, 103, 163, 95, 304} \begin {gather*} \frac {\left (13+7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (13-7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \text {ArcTan}\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} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}-\frac {\left (3-\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}-\sqrt [4]{x^4-x^3} (1-x)+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{4} \sqrt [4]{x^4-x^3}+\frac {\left (13+7 \sqrt {5}\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 {\left (13-7 \sqrt {5}\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 {\left (3+\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}+\frac {\left (3-\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-x^3 + x^4)^(1/4))/4 + ((3 - Sqrt[5])*(-x^3 + x^4)^(1/4))/2 + ((3 + Sqrt[5])*(-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)) + ((1
3 - 7*Sqrt[5])*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((13 + 7*Sqrt[5
])*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((3 + Sqrt[5])^(5/4)*(-x^3
+ x^4)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2^(1/4)*(-1 + x)^(1/4)*x^(3/4)) - ((3
- Sqrt[5])^(5/4)*(-x^3 + x^4)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2^(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)) + ((13 -
7*Sqrt[5])*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((13 + 7*Sqrt[5])*
(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) - ((3 + Sqrt[5])^(5/4)*(-x^3 +
x^4)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2^(1/4)*(-1 + x)^(1/4)*x^(3/4)) + ((3 -
Sqrt[5])^(5/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2^(1/4)*(-1 + x
)^(1/4)*x^(3/4))

Rule 52

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 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 103

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 163

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 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 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 {\left (-1+x+2 x^2\right ) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-1+x+2 x^2\right )}{-1-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 {\sqrt [4]{-1+x} x^{3/4} (1+3 x)}{-1-x+x^2}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} (1+3 x)}{-1-x+x^2} \, 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}}\\ &=-\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 {\sqrt [4]{-x^3+x^4} \int \left (\frac {\left (3+\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1-\sqrt {5}+2 x}+\frac {\left (3-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1+\sqrt {5}+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 (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1+\sqrt {5}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-\sqrt {5}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 ) \text {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 (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1-\sqrt {5}\right )+\frac {1}{2} \left (-1+2 \sqrt {5}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt {5}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1+\sqrt {5}\right )+\frac {1}{2} \left (-1-2 \sqrt {5}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-\sqrt {5}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 ) \text {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 (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt {5}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-\sqrt {5}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\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 (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\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 {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 ) \text {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 ) \text {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 (4 \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1-\sqrt {5}-\left (1-\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 (2 \sqrt {2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt {2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \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 )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\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 {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \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 )}{4 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\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 {\left (13-7 \sqrt {5}\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 {\left (13+7 \sqrt {5}\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 {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\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 {\left (13-7 \sqrt {5}\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 {\left (13+7 \sqrt {5}\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 {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\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 = 1.21, size = 392, normalized size = 1.50 \begin {gather*} \frac {(-1+x)^{3/4} x^{9/4} \left (22 \sqrt [4]{-1+x} x^{3/4}+8 \sqrt [4]{-1+x} x^{7/4}-49 \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+12 \sqrt {2 \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+4 \sqrt {10 \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-12 \sqrt {2 \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+4 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+49 \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-4 \sqrt {2 \left (1+\sqrt {5}\right )} \left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+12 \sqrt {2 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-4 \sqrt {10 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{8 \left ((-1+x) x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)^(3/4)*x^(9/4)*(22*(-1 + x)^(1/4)*x^(3/4) + 8*(-1 + x)^(1/4)*x^(7/4) - 49*ArcTan[((-1 + x)/x)^(-1/4)]
+ 12*Sqrt[2*(1 + Sqrt[5])]*ArcTan[Sqrt[(-1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] + 4*Sqrt[10*(1 + Sqrt[5])]*ArcTa
n[Sqrt[(-1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] - 12*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[Sqrt[(1 + Sqrt[5])/2]/((-1 + x
)/x)^(1/4)] + 4*Sqrt[10*(-1 + Sqrt[5])]*ArcTan[Sqrt[(1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] + 49*ArcTanh[((-1 + x
)/x)^(-1/4)] - 4*Sqrt[2*(1 + Sqrt[5])]*(3 + Sqrt[5])*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] + 12*S
qrt[2*(-1 + Sqrt[5])]*ArcTanh[Sqrt[(1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] - 4*Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[Sq
rt[(1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)]))/(8*((-1 + x)*x^3)^(3/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 19.40, size = 1912, normalized size = 7.33

method	result	size
trager	\(\text {Expression too large to display}\)	\(1912\)
risch	\(\text {Expression too large to display}\)	\(4140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11/4+x)*(x^4-x^3)^(1/4)-49/16*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)
-49/16*RootOf(_Z^2+1)*ln(-(2*RootOf(_Z^2+1)*(x^4-x^3)^(1/2)*x-2*RootOf(_Z^2+1)*x^3+RootOf(_Z^2+1)*x^2-2*(x^4-x
^3)^(3/4)+2*x^2*(x^4-x^3)^(1/4))/x^2)-1/4*RootOf(_Z^4+176*_Z^2-256)*ln(-(RootOf(_Z^4+176*_Z^2-256)^5*x^3-2*Roo
tOf(_Z^4+176*_Z^2-256)^5*x^2+1040*RootOf(_Z^4+176*_Z^2-256)^3*(x^4-x^3)^(1/2)*x-584*RootOf(_Z^4+176*_Z^2-256)^
3*x^3+14400*(x^4-x^3)^(3/4)*RootOf(_Z^4+176*_Z^2-256)^2-9280*(x^4-x^3)^(1/4)*RootOf(_Z^4+176*_Z^2-256)^2*x^2+2
88*RootOf(_Z^4+176*_Z^2-256)^3*x^2+30720*RootOf(_Z^4+176*_Z^2-256)*(x^4-x^3)^(1/2)*x+36864*RootOf(_Z^4+176*_Z^
2-256)*x^3-51200*(x^4-x^3)^(3/4)-35840*x^2*(x^4-x^3)^(1/4)-10368*RootOf(_Z^4+176*_Z^2-256)*x^2)/(x*RootOf(_Z^4
+176*_Z^2-256)^2-2*RootOf(_Z^4+176*_Z^2-256)^2-32*x-16)/x^2)+1/4*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*
ln(-(-RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^4*x^3+2*RootOf(RootOf(_Z^4+176*_Z
^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^4*x^2+1040*(x^4-x^3)^(1/2)*RootOf(_Z^4+176*_Z^2-256)^2*RootOf(Ro
otOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x-936*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-25
6)^2*x^3+992*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^2*x^2-14400*(x^4-x^3)^(3/4
)*RootOf(_Z^4+176*_Z^2-256)^2+9280*(x^4-x^3)^(1/4)*RootOf(_Z^4+176*_Z^2-256)^2*x^2+152320*(x^4-x^3)^(1/2)*Root
Of(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x-170624*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x^3+123008*Root
Of(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x^2-2585600*(x^4-x^3)^(3/4)+1597440*x^2*(x^4-x^3)^(1/4))/(x*RootOf(_Z
^4+176*_Z^2-256)^2-2*RootOf(_Z^4+176*_Z^2-256)^2+208*x-336)/x^2)+1/64*ln((-31*RootOf(_Z^4+176*_Z^2-256)^5*x^3+
62*RootOf(_Z^4+176*_Z^2-256)^5*x^2-19040*RootOf(_Z^4+176*_Z^2-256)^3*(x^4-x^3)^(1/2)*x-26816*RootOf(_Z^4+176*_
Z^2-256)^3*x^3+28800*(x^4-x^3)^(3/4)*RootOf(_Z^4+176*_Z^2-256)^2+18560*(x^4-x^3)^(1/4)*RootOf(_Z^4+176*_Z^2-25
6)^2*x^2+26352*RootOf(_Z^4+176*_Z^2-256)^3*x^2-3384320*RootOf(_Z^4+176*_Z^2-256)*(x^4-x^3)^(1/2)*x-3775744*Roo
tOf(_Z^4+176*_Z^2-256)*x^3+5171200*(x^4-x^3)^(3/4)+3194880*x^2*(x^4-x^3)^(1/4)+2722048*RootOf(_Z^4+176*_Z^2-25
6)*x^2)/(x*RootOf(_Z^4+176*_Z^2-256)^2-2*RootOf(_Z^4+176*_Z^2-256)^2+208*x-336)/x^2)*RootOf(_Z^4+176*_Z^2-256)
^3+11/4*ln((-31*RootOf(_Z^4+176*_Z^2-256)^5*x^3+62*RootOf(_Z^4+176*_Z^2-256)^5*x^2-19040*RootOf(_Z^4+176*_Z^2-
256)^3*(x^4-x^3)^(1/2)*x-26816*RootOf(_Z^4+176*_Z^2-256)^3*x^3+28800*(x^4-x^3)^(3/4)*RootOf(_Z^4+176*_Z^2-256)
^2+18560*(x^4-x^3)^(1/4)*RootOf(_Z^4+176*_Z^2-256)^2*x^2+26352*RootOf(_Z^4+176*_Z^2-256)^3*x^2-3384320*RootOf(
_Z^4+176*_Z^2-256)*(x^4-x^3)^(1/2)*x-3775744*RootOf(_Z^4+176*_Z^2-256)*x^3+5171200*(x^4-x^3)^(3/4)+3194880*x^2
*(x^4-x^3)^(1/4)+2722048*RootOf(_Z^4+176*_Z^2-256)*x^2)/(x*RootOf(_Z^4+176*_Z^2-256)^2-2*RootOf(_Z^4+176*_Z^2-
256)^2+208*x-336)/x^2)*RootOf(_Z^4+176*_Z^2-256)+1/64*RootOf(_Z^4+176*_Z^2-256)^2*RootOf(RootOf(_Z^4+176*_Z^2-
256)^2+_Z^2+176)*ln(-(-31*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^4*x^3+62*Root
Of(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^4*x^2+19040*(x^4-x^3)^(1/2)*RootOf(_Z^4+176
*_Z^2-256)^2*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x+15904*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)
*RootOf(_Z^4+176*_Z^2-256)^2*x^3-4528*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*RootOf(_Z^4+176*_Z^2-256)^2
*x^2+28800*(x^4-x^3)^(3/4)*RootOf(_Z^4+176*_Z^2-256)^2+18560*(x^4-x^3)^(1/4)*RootOf(_Z^4+176*_Z^2-256)^2*x^2-3
3280*(x^4-x^3)^(1/2)*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x-16384*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_
Z^2+176)*x^3+4608*RootOf(RootOf(_Z^4+176*_Z^2-256)^2+_Z^2+176)*x^2-102400*(x^4-x^3)^(3/4)+71680*x^2*(x^4-x^3)^
(1/4))/(x*RootOf(_Z^4+176*_Z^2-256)^2-2*RootOf(_Z^4+176*_Z^2-256)^2-32*x-16)/x^2)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (189) = 378\).
time = 0.53, size = 474, normalized size = 1.82 \begin {gather*} -2 \, \sqrt {10 \, \sqrt {5} - 22} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {5} x + 3 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} \sqrt {\frac {\sqrt {5} x^{2} + x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {10 \, \sqrt {5} - 22} {\left (\sqrt {5} + 3\right )}}{8 \, x}\right ) - 2 \, \sqrt {10 \, \sqrt {5} + 22} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} \sqrt {\frac {\sqrt {5} x^{2} - x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )}}{8 \, x}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (\frac {{\left (\sqrt {5} x - 2 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (-\frac {{\left (\sqrt {5} x - 2 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (\frac {{\left (\sqrt {5} x + 2 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (-\frac {{\left (\sqrt {5} x + 2 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} - 2 \, {\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 {49}{8} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {49}{16} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {49}{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-1)*(x^4-x^3)^(1/4)/(x^2-x-1),x, algorithm="fricas")

[Out]

-2*sqrt(10*sqrt(5) - 22)*arctan(1/8*(sqrt(2)*(sqrt(5)*x + 3*x)*sqrt(10*sqrt(5) - 22)*sqrt((sqrt(5)*x^2 + x^2 +
2*sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(10*sqrt(5) - 22)*(sqrt(5) + 3))/x) - 2*sqrt(10*sqrt(5) + 2
2)*arctan(1/8*(sqrt(2)*(sqrt(5)*x - 3*x)*sqrt(10*sqrt(5) + 22)*sqrt((sqrt(5)*x^2 - x^2 + 2*sqrt(x^4 - x^3))/x^
2) - 2*(x^4 - x^3)^(1/4)*sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3))/x) - 1/2*sqrt(10*sqrt(5) + 22)*log(((sqrt(5)*x -
2*x)*sqrt(10*sqrt(5) + 22) + 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(10*sqrt(5) + 22)*log(-((sqrt(5)*x - 2*x)*sqrt
(10*sqrt(5) + 22) - 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(10*sqrt(5) - 22)*log(((sqrt(5)*x + 2*x)*sqrt(10*sqrt(5)
- 22) + 2*(x^4 - x^3)^(1/4))/x) - 1/2*sqrt(10*sqrt(5) - 22)*log(-((sqrt(5)*x + 2*x)*sqrt(10*sqrt(5) - 22) - 2
*(x^4 - x^3)^(1/4))/x) + 1/4*(x^4 - x^3)^(1/4)*(4*x + 11) + 49/8*arctan((x^4 - x^3)^(1/4)/x) + 49/16*log((x +
(x^4 - x^3)^(1/4))/x) - 49/16*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 )} \left (x + 1\right ) \left (2 x - 1\right )}{x^{2} - x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.51, size = 261, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, {\left (11 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 15 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \sqrt {10 \, \sqrt {5} - 22} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \sqrt {10 \, \sqrt {5} + 22} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {49}{8} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {49}{16} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {49}{16} \, \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((2*x^2+x-1)*(x^4-x^3)^(1/4)/(x^2-x-1),x, algorithm="giac")

[Out]

1/4*(11*(-1/x + 1)^(5/4) - 15*(-1/x + 1)^(1/4))*x^2 - sqrt(10*sqrt(5) - 22)*arctan((-1/x + 1)^(1/4)/sqrt(1/2*s
qrt(5) + 1/2)) + sqrt(10*sqrt(5) + 22)*arctan((-1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/2)) - 1/2*sqrt(10*sqrt(5)
- 22)*log(sqrt(1/2*sqrt(5) + 1/2) + (-1/x + 1)^(1/4)) + 1/2*sqrt(10*sqrt(5) + 22)*log(sqrt(1/2*sqrt(5) - 1/2)
+ (-1/x + 1)^(1/4)) + 1/2*sqrt(10*sqrt(5) - 22)*log(abs(-sqrt(1/2*sqrt(5) + 1/2) + (-1/x + 1)^(1/4))) - 1/2*sq
rt(10*sqrt(5) + 22)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (-1/x + 1)^(1/4))) - 49/8*arctan((-1/x + 1)^(1/4)) - 49
/16*log((-1/x + 1)^(1/4) + 1) + 49/16*log(abs((-1/x + 1)^(1/4) - 1))

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

[Out]

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

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