Integrand size = 24, antiderivative size = 154 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {2}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {2}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (8 \sqrt [4]{2} \left (\arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^(9/4)*(8*2^(1/4)*(ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)] - A rcTanh[2^(1/4)/((-1 + x)/x)^(1/4)]) - RootSum[1 - #1^4 + #1^8 & , (Log[x] - 4*Log[(-1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(-1 + x)^(1/4) - x^(1/4)*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]))/(12*((-1 + x)*x^3)^(3/4))
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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-x^3}}{x \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1}}{\sqrt [4]{x} \left (x^3+1\right )}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}+i\right )}-\frac {i \sqrt [4]{x-1} \sqrt {x}}{2 \left (x^{3/2}-i\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} \sqrt {x}}{x^3+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
3.22.20.3.1 Defintions of rubi rules used
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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 8.57 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}-\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{3}\) | \(122\) |
| trager | \(\text {Expression too large to display}\) | \(3099\) |
-1/3*ln((-2^(1/4)*x-(x^3*(-1+x))^(1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4)))*2^ (1/4)-2/3*arctan(1/2*2^(3/4)/x*(x^3*(-1+x))^(1/4))*2^(1/4)+1/3*sum((_R^4+1 )*ln((-_R*x+(x^3*(-1+x))^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.81 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x + x\right )} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x - x\right )} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x - x\right )} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x + x\right )} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x + x\right )} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x - x\right )} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x - x\right )} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x + x\right )} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{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 {1}{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 {1}{3} 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 {1}{3} 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 ) \]
1/6*sqrt(2)*sqrt(-sqrt(2*I*sqrt(3) + 2))*log((sqrt(2)*(I*sqrt(3)*x + x)*sq rt(-sqrt(2*I*sqrt(3) + 2)) + 4*(x^4 - x^3)^(1/4))/x) - 1/6*sqrt(2)*sqrt(-s qrt(2*I*sqrt(3) + 2))*log((sqrt(2)*(-I*sqrt(3)*x - x)*sqrt(-sqrt(2*I*sqrt( 3) + 2)) + 4*(x^4 - x^3)^(1/4))/x) - 1/6*sqrt(2)*sqrt(-sqrt(-2*I*sqrt(3) + 2))*log((sqrt(2)*(I*sqrt(3)*x - x)*sqrt(-sqrt(-2*I*sqrt(3) + 2)) + 4*(x^4 - x^3)^(1/4))/x) + 1/6*sqrt(2)*sqrt(-sqrt(-2*I*sqrt(3) + 2))*log((sqrt(2) *(-I*sqrt(3)*x + x)*sqrt(-sqrt(-2*I*sqrt(3) + 2)) + 4*(x^4 - x^3)^(1/4))/x ) + 1/6*sqrt(2)*(2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*(I*sqrt(3)*x + x)*(2* I*sqrt(3) + 2)^(1/4) + 4*(x^4 - x^3)^(1/4))/x) - 1/6*sqrt(2)*(2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*(-I*sqrt(3)*x - x)*(2*I*sqrt(3) + 2)^(1/4) + 4*(x^ 4 - x^3)^(1/4))/x) - 1/6*sqrt(2)*(-2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*(I* sqrt(3)*x - x)*(-2*I*sqrt(3) + 2)^(1/4) + 4*(x^4 - x^3)^(1/4))/x) + 1/6*sq rt(2)*(-2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*(-I*sqrt(3)*x + x)*(-2*I*sqrt( 3) + 2)^(1/4) + 4*(x^4 - x^3)^(1/4))/x) - 1/3*2^(1/4)*log((2^(1/4)*x + (x^ 4 - x^3)^(1/4))/x) + 1/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x) - 1/3*I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 1/3*I*2^(1/4)*lo g((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x)
Not integrable
Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{3} + 1\right )} x} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.55 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {2}{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 {1}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]
1/6*(sqrt(6) - sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/(s qrt(6) + sqrt(2))) + 1/6*(sqrt(6) - sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) + 1/6*(sqrt(6) + sqrt(2))*arctan( (sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/6*(sqrt( 6) + sqrt(2))*arctan(-(sqrt(6) + sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/12*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/x + 1 )^(1/4) + sqrt(-1/x + 1) + 1) - 1/12*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 1/12*(sqrt(6) + sqrt( 2))*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1 /12*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sq rt(-1/x + 1) + 1) - 2/3*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) - 1/3 *2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 1/3*2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4)))
Not integrable
Time = 5.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (x^3+1\right )} \,d x \]