Integrand size = 47, antiderivative size = 95 \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (-3+7 x^3+3 x^4\right ) \left (-x+x^5\right )^{3/4}}{21 x^6}-2\ 2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right )-2\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right ) \]
4/21*(3*x^4+7*x^3-3)*(x^5-x)^(3/4)/x^6-2*2^(3/4)*arctan(2^(1/4)*(x^5-x)^(3 /4)/(x^4-1))-2*2^(3/4)*arctanh(2^(1/4)*(x^5-x)^(3/4)/(x^4-1))
\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx \]
Integrate[((-1 + x^4)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-1 - 2*x^3 + x^4)* (-x + x^5)^(1/4)),x]
Integrate[((-1 + x^4)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-1 - 2*x^3 + x^4)* (-x + x^5)^(1/4)), x]
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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \left (x^4+3\right ) \left (x^4-x^3-1\right )}{x^6 \left (x^4-2 x^3-1\right ) \sqrt [4]{x^5-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^4-1} \int \frac {\left (-x^4+x^3+1\right ) \left (x^4-1\right )^{3/4} \left (x^4+3\right )}{x^{25/4} \left (-x^4+2 x^3+1\right )}dx}{\sqrt [4]{x^5-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{x^4-1} \int \frac {\left (-x^4+x^3+1\right ) \left (x^4-1\right )^{3/4} \left (x^4+3\right )}{x^{11/2} \left (-x^4+2 x^3+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^5-x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{x^4-1} \int \left (\frac {2 \sqrt {x} \left (x^4-1\right )^{3/4} (3-2 x)}{-x^4+2 x^3+1}+\frac {\left (x^4-1\right )^{3/4}}{x^{3/2}}-\frac {3 \left (x^4-1\right )^{3/4}}{x^{5/2}}+\frac {3 \left (x^4-1\right )^{3/4}}{x^{11/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{x^4-1} \left (-6 \int \frac {\sqrt {x} \left (x^4-1\right )^{3/4}}{x^4-2 x^3-1}d\sqrt [4]{x}+4 \int \frac {x^{3/2} \left (x^4-1\right )^{3/4}}{x^4-2 x^3-1}d\sqrt [4]{x}-\frac {\left (x^4-1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},-\frac {3}{4},-\frac {5}{16},x^4\right )}{7 x^{21/4} \left (1-x^4\right )^{3/4}}+\frac {\left (x^4-1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {9}{16},\frac {7}{16},x^4\right )}{3 x^{9/4} \left (1-x^4\right )^{3/4}}-\frac {\left (x^4-1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {5}{16},\frac {11}{16},x^4\right )}{5 x^{5/4} \left (1-x^4\right )^{3/4}}\right )}{\sqrt [4]{x^5-x}}\) |
3.14.19.3.1 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]
Time = 19.77 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {\left (12 x^{4}+28 x^{3}-12\right ) \left (x^{5}-x \right )^{\frac {3}{4}}-21 \,2^{\frac {3}{4}} x^{6} \left (\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{5}-x \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{5}-x \right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\left (x^{5}-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )\right )}{21 x^{6}}\) | \(97\) |
| trager | \(\frac {4 \left (3 x^{4}+7 x^{3}-3\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{21 x^{6}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}-2 x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}+x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{5}-x \right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}-2 x^{3}-1}\right )\) | \(275\) |
| risch | \(\frac {\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}+\frac {4}{3} x^{7}-\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{4}-1\right )\right )}^{\frac {1}{4}}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}+x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}-2 x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}-2 x^{3}-1}\right )\) | \(287\) |
1/21*((12*x^4+28*x^3-12)*(x^5-x)^(3/4)-21*2^(3/4)*x^6*(ln((-2^(1/4)*x-(x^5 -x)^(1/4))/(2^(1/4)*x-(x^5-x)^(1/4)))-2*arctan(1/2*(x^5-x)^(1/4)/x*2^(3/4) )))/x^6
Result contains complex when optimal does not.
Time = 65.50 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.74 \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=-\frac {21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) + 21 i \cdot 8^{\frac {1}{4}} x^{6} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + i \cdot 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (-i \, x^{4} - 2 i \, x^{3} + i\right )} - 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 21 i \cdot 8^{\frac {1}{4}} x^{6} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} - i \cdot 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (i \, x^{4} + 2 i \, x^{3} - i\right )} - 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{5} - x} x - 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 8 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} - 3\right )}}{42 \, x^{6}} \]
-1/42*(21*8^(1/4)*x^6*log(-(4*sqrt(2)*(x^5 - x)^(1/4)*x^2 + 8^(3/4)*sqrt(x ^5 - x)*x + 8^(1/4)*(x^4 + 2*x^3 - 1) + 4*(x^5 - x)^(3/4))/(x^4 - 2*x^3 - 1)) + 21*I*8^(1/4)*x^6*log((4*sqrt(2)*(x^5 - x)^(1/4)*x^2 + I*8^(3/4)*sqrt (x^5 - x)*x + 8^(1/4)*(-I*x^4 - 2*I*x^3 + I) - 4*(x^5 - x)^(3/4))/(x^4 - 2 *x^3 - 1)) - 21*I*8^(1/4)*x^6*log((4*sqrt(2)*(x^5 - x)^(1/4)*x^2 - I*8^(3/ 4)*sqrt(x^5 - x)*x + 8^(1/4)*(I*x^4 + 2*I*x^3 - I) - 4*(x^5 - x)^(3/4))/(x ^4 - 2*x^3 - 1)) - 21*8^(1/4)*x^6*log(-(4*sqrt(2)*(x^5 - x)^(1/4)*x^2 - 8^ (3/4)*sqrt(x^5 - x)*x - 8^(1/4)*(x^4 + 2*x^3 - 1) + 4*(x^5 - x)^(3/4))/(x^ 4 - 2*x^3 - 1)) - 8*(x^5 - x)^(3/4)*(3*x^4 + 7*x^3 - 3))/x^6
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (x^4-1\right )\,\left (x^4+3\right )\,\left (-x^4+x^3+1\right )}{x^6\,{\left (x^5-x\right )}^{1/4}\,\left (-x^4+2\,x^3+1\right )} \,d x \]