\(\int \frac {(-1+x+x^4) \sqrt [4]{-x^3+x^4}}{1+x} \, dx\) [1928]

Optimal result

Integrand size = 25, antiderivative size = 134 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {\sqrt [4]{-x^3+x^4} \left (-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right )}{30720}-\frac {9869 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {9869 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \]

[Out]

1/30720*(x^4-x^3)^(1/4)*(6144*x^4-8064*x^3+10400*x^2-1060*x-32575)-9869/4096*arctan(x/(x^4-x^3)^(1/4))+2*2^(1/
4)*arctan(2^(1/4)*x/(x^4-x^3)^(1/4))+9869/4096*arctanh(x/(x^4-x^3)^(1/4))-2*2^(1/4)*arctanh(2^(1/4)*x/(x^4-x^3
)^(1/4))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(134)=268\).

Time = 0.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.55, number of steps used = 18, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {2081, 1629, 161, 96, 95, 304, 209, 212, 963, 81, 52, 65, 246, 218} \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {9869 \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {9869 \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4096 \sqrt [4]{x-1} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {1}{5} x \sqrt [4]{x^4-x^3} (1-x)^3+\frac {27}{80} x \sqrt [4]{x^4-x^3} (1-x)^2-\frac {397}{960} x \sqrt [4]{x^4-x^3} (1-x)-\frac {4 \sqrt [4]{x^4-x^3} (1-x)}{x}-\frac {371 \sqrt [4]{x^4-x^3} (1-x)}{1536}+\frac {4 \sqrt [4]{x^4-x^3}}{x}-\frac {9869 \sqrt [4]{x^4-x^3}}{2048} \]

[In]

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

[Out]

(-9869*(-x^3 + x^4)^(1/4))/2048 - (371*(1 - x)*(-x^3 + x^4)^(1/4))/1536 + (4*(-x^3 + x^4)^(1/4))/x - (4*(1 - x
)*(-x^3 + x^4)^(1/4))/x - (397*(1 - x)*x*(-x^3 + x^4)^(1/4))/960 + (27*(1 - x)^2*x*(-x^3 + x^4)^(1/4))/80 - ((
1 - x)^3*x*(-x^3 + x^4)^(1/4))/5 + (9869*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4096*(-1 + x)^(1/
4)*x^(3/4)) + (2*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4))
+ (9869*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(4096*(-1 + x)^(1/4)*x^(3/4)) - (2*2^(1/4)*(-x^3
+ x^4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 96

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

Rule 161

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol]
:> Dist[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/f^(m + n + 2)), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x
], x] + Dist[1/f^(m + n + 2), Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n
+ 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 1629

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^
(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]

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*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-1+x+x^4\right )}{1+x} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-\frac {13}{4}-\frac {7 x}{4}+\frac {13 x^2}{4}+\frac {27 x^3}{4}\right )}{1+x} \, dx}{5 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-\frac {397}{16}+20 x+\frac {397 x^2}{16}\right )}{1+x} \, dx}{20 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-\frac {1985}{64}+\frac {1855 x}{64}\right )}{1+x} \, dx}{60 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} \left (60-60 x+\frac {1855 x^2}{64}\right )}{x^{5/4}} \, dx}{60 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{x^{5/4} (1+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} \left (-75+\frac {1855 x}{256}\right )}{\sqrt [4]{x}} \, dx}{15 \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} (1+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}-\frac {\left (9869 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{2048 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (8 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {9869 \sqrt [4]{-x^3+x^4}}{2048}-\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {\left (9869 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8192 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {9869 \sqrt [4]{-x^3+x^4}}{2048}-\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9869 \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 )}{2048 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {9869 \sqrt [4]{-x^3+x^4}}{2048}-\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9869 \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 )}{2048 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {9869 \sqrt [4]{-x^3+x^4}}{2048}-\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9869 \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 )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9869 \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 )}{4096 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {9869 \sqrt [4]{-x^3+x^4}}{2048}-\frac {371 (1-x) \sqrt [4]{-x^3+x^4}}{1536}+\frac {4 \sqrt [4]{-x^3+x^4}}{x}-\frac {4 (1-x) \sqrt [4]{-x^3+x^4}}{x}-\frac {397}{960} (1-x) x \sqrt [4]{-x^3+x^4}+\frac {27}{80} (1-x)^2 x \sqrt [4]{-x^3+x^4}-\frac {1}{5} (1-x)^3 x \sqrt [4]{-x^3+x^4}+\frac {9869 \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {9869 \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4096 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {65150 x^3-63030 x^4-22920 x^5+36928 x^6-28416 x^7+12288 x^8-148035 (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+122880 \sqrt [4]{2} (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+148035 (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-122880 \sqrt [4]{2} (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )}{61440 \left ((-1+x) x^3\right )^{3/4}} \]

[In]

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

[Out]

(65150*x^3 - 63030*x^4 - 22920*x^5 + 36928*x^6 - 28416*x^7 + 12288*x^8 - 148035*(-1 + x)^(3/4)*x^(9/4)*ArcTan[
((-1 + x)/x)^(-1/4)] + 122880*2^(1/4)*(-1 + x)^(3/4)*x^(9/4)*ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)] + 148035*(-1 +
x)^(3/4)*x^(9/4)*ArcTanh[((-1 + x)/x)^(-1/4)] - 122880*2^(1/4)*(-1 + x)^(3/4)*x^(9/4)*ArcTanh[2^(1/4)/((-1 +
x)/x)^(1/4)])/(61440*((-1 + x)*x^3)^(3/4))

Maple [A] (verified)

Time = 7.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.49

method	result	size
pseudoelliptic	\(\frac {\left (-5 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-10 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-\frac {49345 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{8192}+\frac {49345 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{8192}+\frac {49345 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{4096}+\left (x^{4}-\frac {21}{16} x^{3}+\frac {325}{192} x^{2}-\frac {265}{1536} x -\frac {32575}{6144}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right ) x^{15}}{5 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{5} {\left (-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+x \right )}^{5} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{5}}\)	\(199\)
trager	\(\left (\frac {1}{5} x^{4}-\frac {21}{80} x^{3}+\frac {65}{192} x^{2}-\frac {53}{1536} x -\frac {6515}{6144}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {9869 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{8192}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )-\frac {9869 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{16384}\)	\(483\)
risch	\(\text {Expression too large to display}\)	\(1040\)

[In]

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

[Out]

1/5*(-5*ln((-2^(1/4)*x-(x^3*(-1+x))^(1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4)))*2^(1/4)-10*arctan(1/2*2^(3/4)/x*(x^
3*(-1+x))^(1/4))*2^(1/4)-49345/8192*ln(((x^3*(-1+x))^(1/4)-x)/x)+49345/8192*ln((x+(x^3*(-1+x))^(1/4))/x)+49345
/4096*arctan((x^3*(-1+x))^(1/4)/x)+(x^4-21/16*x^3+325/192*x^2-265/1536*x-32575/6144)*(x^3*(-1+x))^(1/4))*x^15/
(x+(x^3*(-1+x))^(1/4))^5/(-(x^3*(-1+x))^(1/4)+x)^5/(x^2+(x^3*(-1+x))^(1/2))^5

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.54 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {1}{30720} \, {\left (6144 \, x^{4} - 8064 \, x^{3} + 10400 \, x^{2} - 1060 \, x - 32575\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{4096} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{8192} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {9869}{8192} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

1/30720*(6144*x^4 - 8064*x^3 + 10400*x^2 - 1060*x - 32575)*(x^4 - x^3)^(1/4) - 2^(1/4)*log((2^(1/4)*x + (x^4 -
x^3)^(1/4))/x) + 2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x) - I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(
1/4))/x) + I*2^(1/4)*log((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 9869/4096*arctan((x^4 - x^3)^(1/4)/x) + 9869/
8192*log((x + (x^4 - x^3)^(1/4))/x) - 9869/8192*log(-(x - (x^4 - x^3)^(1/4))/x)

Sympy [F]

\[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{4} + x - 1\right )}{x + 1}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=-\frac {1}{30720} \, {\left (32575 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 131360 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 188230 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 120744 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 25155 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {9869}{4096} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {9869}{8192} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {9869}{8192} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]

[In]

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

[Out]

-1/30720*(32575*(1/x - 1)^4*(-1/x + 1)^(1/4) + 131360*(1/x - 1)^3*(-1/x + 1)^(1/4) + 188230*(1/x - 1)^2*(-1/x
+ 1)^(1/4) - 120744*(-1/x + 1)^(5/4) + 25155*(-1/x + 1)^(1/4))*x^5 - 2*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(
1/4)) - 2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 9869/4096*ar
ctan((-1/x + 1)^(1/4)) + 9869/8192*log((-1/x + 1)^(1/4) + 1) - 9869/8192*log(abs((-1/x + 1)^(1/4) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x^4+x-1\right )}{x+1} \,d x \]

[In]

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

[Out]

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