Integrand size = 29, antiderivative size = 230 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {1}{16} (-5+4 x) \sqrt [4]{-x^3+x^4}-\frac {57}{32} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-x^3+x^4}}{-x^2+\sqrt {-x^3+x^4}}\right )}{12 \sqrt {2}}+\frac {57}{32} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {5 \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x^3+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x^3+x^4}}\right )}{12 \sqrt {2}} \] Output:
1/16*(-5+4*x)*(x^4-x^3)^(1/4)-57/32*arctan(x/(x^4-x^3)^(1/4))+4/3*2^(1/4)* arctan(2^(1/4)*x/(x^4-x^3)^(1/4))+5/24*arctan(2^(1/2)*x*(x^4-x^3)^(1/4)/(- x^2+(x^4-x^3)^(1/2)))*2^(1/2)+57/32*arctanh(x/(x^4-x^3)^(1/4))-4/3*2^(1/4) *arctanh(2^(1/4)*x/(x^4-x^3)^(1/4))-5/24*arctanh((1/2*2^(1/2)*x^2+1/2*(x^4 -x^3)^(1/2)*2^(1/2))/x/(x^4-x^3)^(1/4))*2^(1/2)
Time = 0.74 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (-30 \sqrt [4]{-1+x} x^{3/4}+24 \sqrt [4]{-1+x} x^{7/4}-171 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+128 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+20 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x} \sqrt [4]{x}}{\sqrt {-1+x}-\sqrt {x}}\right )+171 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-128 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-20 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x} \sqrt [4]{x}}{\sqrt {-1+x}+\sqrt {x}}\right )\right )}{96 \left ((-1+x) x^3\right )^{3/4}} \] Input:
Integrate[((1 + x^2)*(-x^3 + x^4)^(1/4))/(-1 + x + 2*x^2),x]
Output:
((-1 + x)^(3/4)*x^(9/4)*(-30*(-1 + x)^(1/4)*x^(3/4) + 24*(-1 + x)^(1/4)*x^ (7/4) - 171*ArcTan[((-1 + x)/x)^(-1/4)] + 128*2^(1/4)*ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)] + 20*Sqrt[2]*ArcTan[(Sqrt[2]*(-1 + x)^(1/4)*x^(1/4))/(Sqrt[ -1 + x] - Sqrt[x])] + 171*ArcTanh[((-1 + x)/x)^(-1/4)] - 128*2^(1/4)*ArcTa nh[2^(1/4)/((-1 + x)/x)^(1/4)] - 20*Sqrt[2]*ArcTanh[(Sqrt[2]*(-1 + x)^(1/4 )*x^(1/4))/(Sqrt[-1 + x] + Sqrt[x])]))/(96*((-1 + x)*x^3)^(3/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 25, 2035, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^2+1\right ) \sqrt [4]{x^4-x^3}}{2 x^2+x-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int -\frac {\sqrt [4]{x-1} x^{3/4} \left (x^2+1\right )}{-2 x^2-x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/4} \left (x^2+1\right )}{-2 x^2-x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/2} \left (x^2+1\right )}{-2 x^2-x+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \left (-\frac {1}{2} \sqrt [4]{x-1} x^{3/2}+\frac {1}{4} \sqrt [4]{x-1} \sqrt {x}+\frac {\sqrt [4]{x-1} (7 x-1) \sqrt {x}}{4 \left (-2 x^2-x+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \left (-\frac {2 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,-x\right )}{9 \sqrt [4]{1-x}}+\frac {5 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,2 x\right )}{36 \sqrt [4]{1-x}}+\frac {1}{128} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )-\frac {1}{128} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )-\frac {1}{16} \sqrt [4]{x-1} x^{7/4}+\frac {5}{64} \sqrt [4]{x-1} x^{3/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
Input:
Int[((1 + x^2)*(-x^3 + x^4)^(1/4))/(-1 + x + 2*x^2),x]
Output:
(-4*(-x^3 + x^4)^(1/4)*((5*(-1 + x)^(1/4)*x^(3/4))/64 - ((-1 + x)^(1/4)*x^ (7/4))/16 - (2*(-1 + x)^(1/4)*x^(3/4)*AppellF1[3/4, -1/4, 1, 7/4, x, -x])/ (9*(1 - x)^(1/4)) + (5*(-1 + x)^(1/4)*x^(3/4)*AppellF1[3/4, -1/4, 1, 7/4, x, 2*x])/(36*(1 - x)^(1/4)) + ArcTan[x^(1/4)/(-1 + x)^(1/4)]/128 - ArcTanh [x^(1/4)/(-1 + x)^(1/4)]/128))/((-1 + x)^(1/4)*x^(3/4))
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.62 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.33
| method | result | size |
| pseudoelliptic | \(\frac {x^{6} \left (48 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} x -128 \ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-256 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-40 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}-40 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}-20 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\sqrt {x^{3} \left (-1+x \right )}-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}}\right ) \sqrt {2}-60 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+171 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )+342 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )-171 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )\right )}{192 \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{2} {\left (\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x \right )}^{2} {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{2}}\) | \(306\) |
| risch | \(\text {Expression too large to display}\) | \(1903\) |
Input:
int((x^2+1)*(x^4-x^3)^(1/4)/(2*x^2+x-1),x,method=_RETURNVERBOSE)
Output:
1/192*x^6*(48*(x^3*(-1+x))^(1/4)*x-128*ln((2^(1/4)*x+(x^3*(-1+x))^(1/4))/( -2^(1/4)*x+(x^3*(-1+x))^(1/4)))*2^(1/4)-256*arctan(1/2*2^(3/4)/x*(x^3*(-1+ x))^(1/4))*2^(1/4)-40*arctan(((x^3*(-1+x))^(1/4)*2^(1/2)+x)/x)*2^(1/2)-40* arctan(((x^3*(-1+x))^(1/4)*2^(1/2)-x)/x)*2^(1/2)-20*ln(((x^3*(-1+x))^(1/4) *2^(1/2)*x+x^2+(x^3*(-1+x))^(1/2))/((x^3*(-1+x))^(1/2)-(x^3*(-1+x))^(1/4)* 2^(1/2)*x+x^2))*2^(1/2)-60*(x^3*(-1+x))^(1/4)+171*ln((x+(x^3*(-1+x))^(1/4) )/x)+342*arctan((x^3*(-1+x))^(1/4)/x)-171*ln(((x^3*(-1+x))^(1/4)-x)/x))/(x ^2+(x^3*(-1+x))^(1/2))^2/((x^3*(-1+x))^(1/4)-x)^2/(x+(x^3*(-1+x))^(1/4))^2
Time = 0.11 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.37 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=-\frac {5}{24} \, \sqrt {2} \arctan \left (\frac {x + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {5}{24} \, \sqrt {2} \arctan \left (-\frac {x - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\frac {x^{2} + \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\frac {x^{2} - \sqrt {2} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x + \sqrt {x^{4} - x^{3}}}{x^{2}}\right ) + \frac {1}{16} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 5\right )} + \frac {4}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{x^{3} - x^{2}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {57}{32} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{x^{3} - x^{2}}\right ) + \frac {57}{64} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {57}{64} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \] Input:
integrate((x^2+1)*(x^4-x^3)^(1/4)/(2*x^2+x-1),x, algorithm="fricas")
Output:
-5/24*sqrt(2)*arctan((x + sqrt(2)*(x^4 - x^3)^(1/4))/x) - 5/24*sqrt(2)*arc tan(-(x - sqrt(2)*(x^4 - x^3)^(1/4))/x) - 5/48*sqrt(2)*log((x^2 + sqrt(2)* (x^4 - x^3)^(1/4)*x + sqrt(x^4 - x^3))/x^2) + 5/48*sqrt(2)*log((x^2 - sqrt (2)*(x^4 - x^3)^(1/4)*x + sqrt(x^4 - x^3))/x^2) + 1/16*(x^4 - x^3)^(1/4)*( 4*x - 5) + 4/3*2^(1/4)*arctan(2^(1/4)*(x^4 - x^3)^(3/4)/(x^3 - x^2)) - 2/3 *2^(1/4)*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2/3*2^(1/4)*log(-(2^(1/4 )*x - (x^4 - x^3)^(1/4))/x) - 57/32*arctan((x^4 - x^3)^(3/4)/(x^3 - x^2)) + 57/64*log((x + (x^4 - x^3)^(1/4))/x) - 57/64*log(-(x - (x^4 - x^3)^(1/4) )/x)
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{2} + 1\right )}{\left (x + 1\right ) \left (2 x - 1\right )}\, dx \] Input:
integrate((x**2+1)*(x**4-x**3)**(1/4)/(2*x**2+x-1),x)
Output:
Integral((x**3*(x - 1))**(1/4)*(x**2 + 1)/((x + 1)*(2*x - 1)), x)
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{2 \, x^{2} + x - 1} \,d x } \] Input:
integrate((x^2+1)*(x^4-x^3)^(1/4)/(2*x^2+x-1),x, algorithm="maxima")
Output:
integrate((x^4 - x^3)^(1/4)*(x^2 + 1)/(2*x^2 + x - 1), x)
Time = 0.58 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {1}{16} \, {\left (5 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {5}{24} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {5}{24} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {5}{48} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {4}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {57}{32} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {57}{64} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {57}{64} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:
integrate((x^2+1)*(x^4-x^3)^(1/4)/(2*x^2+x-1),x, algorithm="giac")
Output:
1/16*(5*(-1/x + 1)^(5/4) - (-1/x + 1)^(1/4))*x^2 - 5/24*sqrt(2)*arctan(1/2 *sqrt(2)*(sqrt(2) + 2*(-1/x + 1)^(1/4))) - 5/24*sqrt(2)*arctan(-1/2*sqrt(2 )*(sqrt(2) - 2*(-1/x + 1)^(1/4))) - 5/48*sqrt(2)*log(sqrt(2)*(-1/x + 1)^(1 /4) + sqrt(-1/x + 1) + 1) + 5/48*sqrt(2)*log(-sqrt(2)*(-1/x + 1)^(1/4) + s qrt(-1/x + 1) + 1) - 4/3*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) - 2/ 3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 2/3*2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 57/32*arctan((-1/x + 1)^(1/4)) + 57/64*log((-1/x + 1 )^(1/4) + 1) - 57/64*log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int \frac {\left (x^2+1\right )\,{\left (x^4-x^3\right )}^{1/4}}{2\,x^2+x-1} \,d x \] Input:
int(((x^2 + 1)*(x^4 - x^3)^(1/4))/(x + 2*x^2 - 1),x)
Output:
int(((x^2 + 1)*(x^4 - x^3)^(1/4))/(x + 2*x^2 - 1), x)
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {x^{\frac {7}{4}} \left (x -1\right )^{\frac {1}{4}}}{4}-\frac {5 x^{\frac {3}{4}} \left (x -1\right )^{\frac {1}{4}}}{16}+\frac {15 \left (\int \frac {\left (x -1\right )^{\frac {1}{4}}}{2 x^{\frac {13}{4}}-x^{\frac {9}{4}}-2 x^{\frac {5}{4}}+x^{\frac {1}{4}}}d x \right )}{64}+\frac {57 \left (\int \frac {\left (x -1\right )^{\frac {1}{4}} x^{2}}{2 x^{\frac {13}{4}}-x^{\frac {9}{4}}-2 x^{\frac {5}{4}}+x^{\frac {1}{4}}}d x \right )}{32}-\frac {127 \left (\int \frac {\left (x -1\right )^{\frac {1}{4}} x}{2 x^{\frac {13}{4}}-x^{\frac {9}{4}}-2 x^{\frac {5}{4}}+x^{\frac {1}{4}}}d x \right )}{64} \] Input:
int((x^2+1)*(x^4-x^3)^(1/4)/(2*x^2+x-1),x)
Output:
(16*x**(3/4)*(x - 1)**(1/4)*x - 20*x**(3/4)*(x - 1)**(1/4) + 15*int((x - 1 )**(1/4)/(2*x**(1/4)*x**3 - x**(1/4)*x**2 - 2*x**(1/4)*x + x**(1/4)),x) + 114*int(((x - 1)**(1/4)*x**2)/(2*x**(1/4)*x**3 - x**(1/4)*x**2 - 2*x**(1/4 )*x + x**(1/4)),x) - 127*int(((x - 1)**(1/4)*x)/(2*x**(1/4)*x**3 - x**(1/4 )*x**2 - 2*x**(1/4)*x + x**(1/4)),x))/64