3.28.89 \(\int \frac {(-x+x^2) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx\) [2789]

Optimal. Leaf size=269 \[ \frac {1}{8} (-1+4 x) \sqrt [4]{-x^3+x^4}-\frac {29}{16} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {29}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right ) \]

[Out]

1/8*(-1+4*x)*(x^4-x^3)^(1/4)-29/16*arctan(x/(x^4-x^3)^(1/4))+1/5*(10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2
))^(1/2)*x/(x^4-x^3)^(1/4))+1/5*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))+29/16
*arctanh(x/(x^4-x^3)^(1/4))-1/5*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))-1/5*
(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^3)^(1/4))

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 436, normalized size of antiderivative = 1.62, number of steps used = 26, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {1607, 2081, 917, 52, 65, 246, 218, 212, 209, 919, 925, 95, 304} \begin {gather*} \frac {29 \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{x-1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \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 {5} \sqrt [4]{x-1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \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 {5} \sqrt [4]{x-1} x^{3/4}}-\frac {1}{2} \sqrt [4]{x^4-x^3} (1-x)+\frac {3}{8} \sqrt [4]{x^4-x^3}+\frac {29 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{x-1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \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 {5} \sqrt [4]{x-1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \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 {5} \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-x^3 + x^4)^(1/4))/8 - ((1 - x)*(-x^3 + x^4)^(1/4))/2 + (29*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/
4)])/(16*(-1 + x)^(1/4)*x^(3/4)) + (2^(3/4)*(3 + Sqrt[5])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(
1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[5]*(-1 + x)^(1/4)*x^(3/4)) + (2^(3/4)*(3 - Sqrt[5])^(1/4)*(-x^3 + x^4)^(1
/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[5]*(-1 + x)^(1/4)*x^(3/4)) + (29*(-x^3 + x
^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(16*(-1 + x)^(1/4)*x^(3/4)) - (2^(3/4)*(3 + Sqrt[5])^(1/4)*(-x^3 +
x^4)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[5]*(-1 + x)^(1/4)*x^(3/4)) - (2^(3
/4)*(3 - Sqrt[5])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(Sqrt[5]
*(-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 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 917

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

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 925

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rubi steps

\begin {align*} \int \frac {\left (-x+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx &=\int \frac {(-1+x) x \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {(-1+x)^{5/4} x^{7/4}}{-1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \sqrt [4]{-1+x} x^{3/4} \, 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-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {1}{2} (1-x) \sqrt [4]{-x^3+x^4}+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{8 \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}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-x+x^2\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{8} \sqrt [4]{-x^3+x^4}-\frac {1}{2} (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}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \left (-\frac {2}{\sqrt {5} \left (1+\sqrt {5}-2 x\right ) (-1+x)^{3/4} \sqrt [4]{x}}-\frac {2}{\sqrt {5} (-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt {5}+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 {3}{8} \sqrt [4]{-x^3+x^4}-\frac {1}{2} (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 )}{8 \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 (2 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{\left (1+\sqrt {5}-2 x\right ) (-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\sqrt {5} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \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 {5} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{8} \sqrt [4]{-x^3+x^4}-\frac {1}{2} (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 )}{8 \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 (8 \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 {5} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (8 \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 {5} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{8} \sqrt [4]{-x^3+x^4}-\frac {1}{2} (1-x) \sqrt [4]{-x^3+x^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 (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 )}{16 \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 )}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt {\frac {2}{5}} \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 (4 \sqrt {\frac {2}{5}} \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 (4 \sqrt {\frac {2}{5}} \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 (4 \sqrt {\frac {2}{5}} \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 {3}{8} \sqrt [4]{-x^3+x^4}-\frac {1}{2} (1-x) \sqrt [4]{-x^3+x^4}+\frac {29 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3+\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 {5} \sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3-\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 {5} \sqrt [4]{-1+x} x^{3/4}}+\frac {29 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3+\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 {5} \sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3-\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 {5} \sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.67, size = 256, normalized size = 0.95 \begin {gather*} \frac {(-1+x)^{3/4} x^{9/4} \left (-10 \sqrt [4]{-1+x} x^{3/4}+40 \sqrt [4]{-1+x} x^{7/4}-145 \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+16 \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 )+16 \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 )+145 \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-16 \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 )-16 \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 )}{80 \left ((-1+x) x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)^(3/4)*x^(9/4)*(-10*(-1 + x)^(1/4)*x^(3/4) + 40*(-1 + x)^(1/4)*x^(7/4) - 145*ArcTan[((-1 + x)/x)^(-1/
4)] + 16*Sqrt[10*(1 + Sqrt[5])]*ArcTan[Sqrt[(-1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] + 16*Sqrt[10*(-1 + Sqrt[5])]
*ArcTan[Sqrt[(1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] + 145*ArcTanh[((-1 + x)/x)^(-1/4)] - 16*Sqrt[10*(1 + Sqrt[5]
)]*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]/((-1 + x)/x)^(1/4)] - 16*Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[Sqrt[(1 + Sqrt[5])/
2]/((-1 + x)/x)^(1/4)]))/(80*((-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 = 21.93, size = 2229, normalized size = 8.29

method	result	size
trager	\(\text {Expression too large to display}\)	\(2229\)
risch	\(\text {Expression too large to display}\)	\(4529\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/8+1/2*x)*(x^4-x^3)^(1/4)+1/40*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*ln(-(425*RootOf(_Z^2+25*
RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*RootOf(25*_Z^4-320*_Z^2-4096)^4*x^3-850*RootOf(_Z^2+25*RootOf(25*_Z^4-320
*_Z^2-4096)^2-320)*RootOf(25*_Z^4-320*_Z^2-4096)^4*x^2-39040*(x^4-x^3)^(1/2)*RootOf(25*_Z^4-320*_Z^2-4096)^2*R
ootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x-51840*RootOf(25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^2+25*Root
Of(25*_Z^4-320*_Z^2-4096)^2-320)*x^3+43840*RootOf(25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_
Z^2-4096)^2-320)*x^2+1152000*(x^4-x^3)^(3/4)*RootOf(25*_Z^4-320*_Z^2-4096)^2+742400*RootOf(25*_Z^4-320*_Z^2-40
96)^2*(x^4-x^3)^(1/4)*x^2+843776*(x^4-x^3)^(1/2)*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x+851968*
RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x^3-532480*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-
320)*x^2-24248320*(x^4-x^3)^(3/4)-14745600*x^2*(x^4-x^3)^(1/4))/(5*x*RootOf(25*_Z^4-320*_Z^2-4096)^2-10*RootOf
(25*_Z^4-320*_Z^2-4096)^2-128*x+192)/x^2)-1/8*RootOf(25*_Z^4-320*_Z^2-4096)*ln((425*x^3*RootOf(25*_Z^4-320*_Z^
2-4096)^5-850*x^2*RootOf(25*_Z^4-320*_Z^2-4096)^5+39040*RootOf(25*_Z^4-320*_Z^2-4096)^3*(x^4-x^3)^(1/2)*x+4096
0*x^3*RootOf(25*_Z^4-320*_Z^2-4096)^3-22080*x^2*RootOf(25*_Z^4-320*_Z^2-4096)^3+230400*(x^4-x^3)^(3/4)*RootOf(
25*_Z^4-320*_Z^2-4096)^2+148480*RootOf(25*_Z^4-320*_Z^2-4096)^2*(x^4-x^3)^(1/4)*x^2+344064*RootOf(25*_Z^4-320*
_Z^2-4096)*(x^4-x^3)^(1/2)*x+258048*RootOf(25*_Z^4-320*_Z^2-4096)*x^3-110592*x^2*RootOf(25*_Z^4-320*_Z^2-4096)
+1900544*(x^4-x^3)^(3/4)+1048576*x^2*(x^4-x^3)^(1/4))/(5*x*RootOf(25*_Z^4-320*_Z^2-4096)^2-10*RootOf(25*_Z^4-3
20*_Z^2-4096)^2+64*x-64)/x^2)+29/32*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)-29/2048*RootOf(25*_Z^4-320*_Z^2-4096)*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*ln(-(2*(x^4-x^
3)^(1/2)*RootOf(25*_Z^4-320*_Z^2-4096)*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x-2*RootOf(25*_Z^4-
320*_Z^2-4096)*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x^3+RootOf(25*_Z^4-320*_Z^2-4096)*RootOf(_Z
^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x^2-128*(x^4-x^3)^(3/4)+128*x^2*(x^4-x^3)^(1/4))/x^2)-1/512*RootOf(
25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*ln(-(325*RootOf(_Z^2+25*RootOf(25
*_Z^4-320*_Z^2-4096)^2-320)*RootOf(25*_Z^4-320*_Z^2-4096)^4*x^3-650*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-409
6)^2-320)*RootOf(25*_Z^4-320*_Z^2-4096)^4*x^2-32960*(x^4-x^3)^(1/2)*RootOf(25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^
2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x+31840*RootOf(25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^2+25*RootOf(25*_Z^
4-320*_Z^2-4096)^2-320)*x^3-17920*RootOf(25*_Z^4-320*_Z^2-4096)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)
^2-320)*x^2-576000*(x^4-x^3)^(3/4)*RootOf(25*_Z^4-320*_Z^2-4096)^2+371200*RootOf(25*_Z^4-320*_Z^2-4096)^2*(x^4
-x^3)^(1/4)*x^2-249856*(x^4-x^3)^(1/2)*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x+243712*RootOf(_Z^
2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x^3-104448*RootOf(_Z^2+25*RootOf(25*_Z^4-320*_Z^2-4096)^2-320)*x^2-4
751360*(x^4-x^3)^(3/4)+2621440*x^2*(x^4-x^3)^(1/4))/(5*x*RootOf(25*_Z^4-320*_Z^2-4096)^2-10*RootOf(25*_Z^4-320
*_Z^2-4096)^2+64*x-64)/x^2)-5/512*ln((325*x^3*RootOf(25*_Z^4-320*_Z^2-4096)^5-650*x^2*RootOf(25*_Z^4-320*_Z^2-
4096)^5+32960*RootOf(25*_Z^4-320*_Z^2-4096)^3*(x^4-x^3)^(1/2)*x-40160*x^3*RootOf(25*_Z^4-320*_Z^2-4096)^3+3456
0*x^2*RootOf(25*_Z^4-320*_Z^2-4096)^3-115200*(x^4-x^3)^(3/4)*RootOf(25*_Z^4-320*_Z^2-4096)^2+74240*RootOf(25*_
Z^4-320*_Z^2-4096)^2*(x^4-x^3)^(1/4)*x^2-671744*RootOf(25*_Z^4-320*_Z^2-4096)*(x^4-x^3)^(1/2)*x+704512*RootOf(
25*_Z^4-320*_Z^2-4096)*x^3-440320*x^2*RootOf(25*_Z^4-320*_Z^2-4096)+2424832*(x^4-x^3)^(3/4)-1474560*x^2*(x^4-x
^3)^(1/4))/(5*x*RootOf(25*_Z^4-320*_Z^2-4096)^2-10*RootOf(25*_Z^4-320*_Z^2-4096)^2-128*x+192)/x^2)*RootOf(25*_
Z^4-320*_Z^2-4096)^3+1/8*ln((325*x^3*RootOf(25*_Z^4-320*_Z^2-4096)^5-650*x^2*RootOf(25*_Z^4-320*_Z^2-4096)^5+3
2960*RootOf(25*_Z^4-320*_Z^2-4096)^3*(x^4-x^3)^(1/2)*x-40160*x^3*RootOf(25*_Z^4-320*_Z^2-4096)^3+34560*x^2*Roo
tOf(25*_Z^4-320*_Z^2-4096)^3-115200*(x^4-x^3)^(3/4)*RootOf(25*_Z^4-320*_Z^2-4096)^2+74240*RootOf(25*_Z^4-320*_
Z^2-4096)^2*(x^4-x^3)^(1/4)*x^2-671744*RootOf(25*_Z^4-320*_Z^2-4096)*(x^4-x^3)^(1/2)*x+704512*RootOf(25*_Z^4-3
20*_Z^2-4096)*x^3-440320*x^2*RootOf(25*_Z^4-320*_Z^2-4096)+2424832*(x^4-x^3)^(3/4)-1474560*x^2*(x^4-x^3)^(1/4)
)/(5*x*RootOf(25*_Z^4-320*_Z^2-4096)^2-10*RootOf(25*_Z^4-320*_Z^2-4096)^2-128*x+192)/x^2)*RootOf(25*_Z^4-320*_
Z^2-4096)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (191) = 382\).
time = 0.41, size = 462, normalized size = 1.72 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {\sqrt {2} x \sqrt {2 \, \sqrt {5} - 2} \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 {2 \, \sqrt {5} - 2}}{4 \, x}\right ) + \frac {2}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} x \sqrt {2 \, \sqrt {5} + 2} \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 {2 \, \sqrt {5} + 2}}{4 \, x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 1\right )} + \frac {29}{16} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {29}{32} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {29}{32} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(5)*sqrt(2*sqrt(5) - 2)*arctan(1/4*(sqrt(2)*x*sqrt(2*sqrt(5) - 2)*sqrt((sqrt(5)*x^2 + x^2 + 2*sqrt(x^4
- x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(2*sqrt(5) - 2))/x) + 2/5*sqrt(5)*sqrt(2*sqrt(5) + 2)*arctan(1/4*(sqrt
(2)*x*sqrt(2*sqrt(5) + 2)*sqrt((sqrt(5)*x^2 - x^2 + 2*sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(2*sqrt(
5) + 2))/x) - 1/10*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(((sqrt(5)*x - x)*sqrt(2*sqrt(5) + 2) + 4*(x^4 - x^3)^(1/4))
/x) + 1/10*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-((sqrt(5)*x - x)*sqrt(2*sqrt(5) + 2) - 4*(x^4 - x^3)^(1/4))/x) - 1
/10*sqrt(5)*sqrt(2*sqrt(5) - 2)*log(((sqrt(5)*x + x)*sqrt(2*sqrt(5) - 2) + 4*(x^4 - x^3)^(1/4))/x) + 1/10*sqrt
(5)*sqrt(2*sqrt(5) - 2)*log(-((sqrt(5)*x + x)*sqrt(2*sqrt(5) - 2) - 4*(x^4 - x^3)^(1/4))/x) + 1/8*(x^4 - x^3)^
(1/4)*(4*x - 1) + 29/16*arctan((x^4 - x^3)^(1/4)/x) + 29/32*log((x + (x^4 - x^3)^(1/4))/x) - 29/32*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 {x \sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right )}{x^{2} - x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.50, size = 260, normalized size = 0.97 \begin {gather*} -\frac {1}{8} \, {\left ({\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} + \frac {1}{5} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{5} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \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}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {29}{16} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {29}{32} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {29}{32} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((-1/x + 1)^(5/4) + 3*(-1/x + 1)^(1/4))*x^2 + 1/5*sqrt(10*sqrt(5) - 10)*arctan((-1/x + 1)^(1/4)/sqrt(1/2*
sqrt(5) + 1/2)) + 1/5*sqrt(10*sqrt(5) + 10)*arctan((-1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/2)) + 1/10*sqrt(10*sq
rt(5) - 10)*log(sqrt(1/2*sqrt(5) + 1/2) + (-1/x + 1)^(1/4)) + 1/10*sqrt(10*sqrt(5) + 10)*log(sqrt(1/2*sqrt(5)
- 1/2) + (-1/x + 1)^(1/4)) - 1/10*sqrt(10*sqrt(5) - 10)*log(abs(-sqrt(1/2*sqrt(5) + 1/2) + (-1/x + 1)^(1/4)))
- 1/10*sqrt(10*sqrt(5) + 10)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (-1/x + 1)^(1/4))) - 29/16*arctan((-1/x + 1)^(
1/4)) - 29/32*log((-1/x + 1)^(1/4) + 1) + 29/32*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-x^2\right )\,{\left (x^4-x^3\right )}^{1/4}}{-x^2+x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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