\(\int \frac {(-1+x^4) (3+x^4) (-1-x^3+x^4)}{x^6 (-1-2 x^3+x^4) \sqrt [4]{-x+x^5}} \, dx\) [1319]

Optimal result

Integrand size = 47, antiderivative size = 95 \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (-3+7 x^3+3 x^4\right ) \left (-x+x^5\right )^{3/4}}{21 x^6}-2\ 2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right )-2\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right ) \]

[Out]

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))

Rubi [F]

\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx \]

[In]

Int[((-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]

[Out]

(4*(1 - x^4)^(1/4)*Hypergeometric2F1[-21/16, -3/4, -5/16, x^4])/(7*x^5*(-x + x^5)^(1/4)) - (4*(1 - x^4)^(1/4)*
Hypergeometric2F1[-3/4, -9/16, 7/16, x^4])/(3*x^2*(-x + x^5)^(1/4)) + (4*(1 - x^4)^(1/4)*Hypergeometric2F1[-3/
4, -5/16, 11/16, x^4])/(5*x*(-x + x^5)^(1/4)) - (24*x^(1/4)*(-1 + x^4)^(1/4)*Defer[Subst][Defer[Int][(x^2*(-1
+ x^16)^(3/4))/(-1 - 2*x^12 + x^16), x], x, x^(1/4)])/(-x + x^5)^(1/4) + (16*x^(1/4)*(-1 + x^4)^(1/4)*Defer[Su
bst][Defer[Int][(x^6*(-1 + x^16)^(3/4))/(-1 - 2*x^12 + x^16), x], x, x^(1/4)])/(-x + x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {\left (-1+x^4\right )^{3/4} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^{25/4} \left (-1-2 x^3+x^4\right )} \, dx}{\sqrt [4]{-x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4} \left (3+x^{16}\right ) \left (-1-x^{12}+x^{16}\right )}{x^{22} \left (-1-2 x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \left (\frac {3 \left (-1+x^{16}\right )^{3/4}}{x^{22}}-\frac {3 \left (-1+x^{16}\right )^{3/4}}{x^{10}}+\frac {\left (-1+x^{16}\right )^{3/4}}{x^6}+\frac {2 x^2 \left (3-2 x^4\right ) \left (-1+x^{16}\right )^{3/4}}{1+2 x^{12}-x^{16}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^6} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (3-2 x^4\right ) \left (-1+x^{16}\right )^{3/4}}{1+2 x^{12}-x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (12 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^{10}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {3 x^2 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}}+\frac {2 x^6 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (4 \sqrt [4]{x} \left (-1+x^4\right )\right ) \text {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^6} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}}+\frac {\left (12 \sqrt [4]{x} \left (-1+x^4\right )\right ) \text {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}}-\frac {\left (12 \sqrt [4]{x} \left (-1+x^4\right )\right ) \text {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^{10}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}} \\ & = \frac {4 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},-\frac {3}{4},-\frac {5}{16},x^4\right )}{7 x^5 \sqrt [4]{-x+x^5}}-\frac {4 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {9}{16},\frac {7}{16},x^4\right )}{3 x^2 \sqrt [4]{-x+x^5}}+\frac {4 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {5}{16},\frac {11}{16},x^4\right )}{5 x \sqrt [4]{-x+x^5}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}-\frac {\left (24 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx \]

[In]

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]

[Out]

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]

Maple [A] (verified)

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	\(\frac {\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}+\frac {4}{3} x^{7}-\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{4}-1\right )\right )}^{\frac {1}{4}}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}+x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}-2 x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}-2 x^{3}-1}\right )\)	\(287\)

[In]

int((x^4-1)*(x^4+3)*(x^4-x^3-1)/x^6/(x^4-2*x^3-1)/(x^5-x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 65.50 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.74 \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=-\frac {21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) + 21 i \cdot 8^{\frac {1}{4}} x^{6} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + i \cdot 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (-i \, x^{4} - 2 i \, x^{3} + i\right )} - 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 21 i \cdot 8^{\frac {1}{4}} x^{6} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} - i \cdot 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (i \, x^{4} + 2 i \, x^{3} - i\right )} - 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{5} - x} x - 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 8 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} - 3\right )}}{42 \, x^{6}} \]

[In]

integrate((x^4-1)*(x^4+3)*(x^4-x^3-1)/x^6/(x^4-2*x^3-1)/(x^5-x)^(1/4),x, algorithm="fricas")

[Out]

-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)*sq
rt(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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\text {Timed out} \]

[In]

integrate((x**4-1)*(x**4+3)*(x**4-x**3-1)/x**6/(x**4-2*x**3-1)/(x**5-x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-1)*(x^4+3)*(x^4-x^3-1)/x^6/(x^4-2*x^3-1)/(x^5-x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 - 1)*(x^4 + 3)*(x^4 - 1)/((x^5 - x)^(1/4)*(x^4 - 2*x^3 - 1)*x^6), x)

Giac [F]

\[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-1)*(x^4+3)*(x^4-x^3-1)/x^6/(x^4-2*x^3-1)/(x^5-x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 - 1)*(x^4 + 3)*(x^4 - 1)/((x^5 - x)^(1/4)*(x^4 - 2*x^3 - 1)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (x^4-1\right )\,\left (x^4+3\right )\,\left (-x^4+x^3+1\right )}{x^6\,{\left (x^5-x\right )}^{1/4}\,\left (-x^4+2\,x^3+1\right )} \,d x \]

[In]

int(((x^4 - 1)*(x^4 + 3)*(x^3 - x^4 + 1))/(x^6*(x^5 - x)^(1/4)*(2*x^3 - x^4 + 1)),x)

[Out]

int(((x^4 - 1)*(x^4 + 3)*(x^3 - x^4 + 1))/(x^6*(x^5 - x)^(1/4)*(2*x^3 - x^4 + 1)), x)