\(\int \frac {(4+3 x) (-1-x+x^4) \sqrt [4]{-1-x+2 x^4}}{x^6 (1+x+x^4)} \, dx\) [1238]

Optimal result

Integrand size = 39, antiderivative size = 90 \[ \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx=-\frac {4 \sqrt [4]{-1-x+2 x^4} \left (-1-x+12 x^4\right )}{5 x^5}-4 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right )+4 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right ) \]

[Out]

-4/5*(2*x^4-x-1)^(1/4)*(12*x^4-x-1)/x^5-4*3^(1/4)*arctan(3^(1/4)*x/(2*x^4-x-1)^(1/4))+4*3^(1/4)*arctanh(3^(1/4
)*x/(2*x^4-x-1)^(1/4))

Rubi [F]

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

[In]

Int[((4 + 3*x)*(-1 - x + x^4)*(-1 - x + 2*x^4)^(1/4))/(x^6*(1 + x + x^4)),x]

[Out]

-4*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x^6, x] - 3*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x^5, x] + 8*Defer[Int][(-1
- x + 2*x^4)^(1/4)/x^2, x] - 2*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x, x] + 2*Defer[Int][(-1 - x + 2*x^4)^(1/4)/(
1 + x + x^4), x] - 8*Defer[Int][(x^2*(-1 - x + 2*x^4)^(1/4))/(1 + x + x^4), x] + 2*Defer[Int][(x^3*(-1 - x + 2
*x^4)^(1/4))/(1 + x + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4 \sqrt [4]{-1-x+2 x^4}}{x^6}-\frac {3 \sqrt [4]{-1-x+2 x^4}}{x^5}+\frac {8 \sqrt [4]{-1-x+2 x^4}}{x^2}-\frac {2 \sqrt [4]{-1-x+2 x^4}}{x}+\frac {2 \left (1-4 x^2+x^3\right ) \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \frac {\left (1-4 x^2+x^3\right ) \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \left (\frac {\sqrt [4]{-1-x+2 x^4}}{1+x+x^4}-\frac {4 x^2 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}+\frac {x^3 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}\right ) \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx+2 \int \frac {x^3 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx-8 \int \frac {x^2 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx=-\frac {4 \sqrt [4]{-1-x+2 x^4} \left (-1-x+12 x^4\right )}{5 x^5}-4 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right )+4 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right ) \]

[In]

Integrate[((4 + 3*x)*(-1 - x + x^4)*(-1 - x + 2*x^4)^(1/4))/(x^6*(1 + x + x^4)),x]

[Out]

(-4*(-1 - x + 2*x^4)^(1/4)*(-1 - x + 12*x^4))/(5*x^5) - 4*3^(1/4)*ArcTan[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)] +
4*3^(1/4)*ArcTanh[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)]

Maple [A] (verified)

Time = 14.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {2 x^{5} \left (\ln \left (\frac {3^{\frac {1}{4}} x +\left (2 x^{4}-x -1\right )^{\frac {1}{4}}}{-3^{\frac {1}{4}} x +\left (2 x^{4}-x -1\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \left (2 x^{4}-x -1\right )^{\frac {1}{4}}}{3 x}\right )\right ) 3^{\frac {1}{4}}-\frac {4 \left (2 x^{4}-x -1\right )^{\frac {1}{4}} \left (12 x^{4}-x -1\right )}{5}}{x^{5}}\)	\(104\)
trager	\(-\frac {4 \left (2 x^{4}-x -1\right )^{\frac {1}{4}} \left (12 x^{4}-x -1\right )}{5 x^{5}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3} x^{4}+6 \left (2 x^{4}-x -1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {2 x^{4}-x -1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right ) x^{2}+6 \left (2 x^{4}-x -1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3}}{x^{4}+x +1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{4}-6 \left (2 x^{4}-x -1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {2 x^{4}-x -1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{2}+6 \left (2 x^{4}-x -1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right )}{x^{4}+x +1}\right )\)	\(311\)
risch	\(\text {Expression too large to display}\)	\(1633\)

[In]

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

[Out]

2/5*(5*x^5*(ln((3^(1/4)*x+(2*x^4-x-1)^(1/4))/(-3^(1/4)*x+(2*x^4-x-1)^(1/4)))+2*arctan(1/3*3^(3/4)/x*(2*x^4-x-1
)^(1/4)))*3^(1/4)-2*(2*x^4-x-1)^(1/4)*(12*x^4-x-1))/x^5

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.39 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.30 \[ \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx=\frac {5 \cdot 3^{\frac {1}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} + 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 1\right )} + 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) - 5 i \cdot 3^{\frac {1}{4}} x^{5} \log \left (-\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} + 6 i \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} - 3^{\frac {3}{4}} {\left (5 i \, x^{4} - i \, x - i\right )} - 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) + 5 i \cdot 3^{\frac {1}{4}} x^{5} \log \left (-\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} - 6 i \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} - 3^{\frac {3}{4}} {\left (-5 i \, x^{4} + i \, x + i\right )} - 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) - 5 \cdot 3^{\frac {1}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} - 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 1\right )} + 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) - 4 \, {\left (12 \, x^{4} - x - 1\right )} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]

[In]

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

[Out]

1/5*(5*3^(1/4)*x^5*log((6*sqrt(3)*(2*x^4 - x - 1)^(1/4)*x^3 + 6*3^(1/4)*sqrt(2*x^4 - x - 1)*x^2 + 3^(3/4)*(5*x
^4 - x - 1) + 6*(2*x^4 - x - 1)^(3/4)*x)/(x^4 + x + 1)) - 5*I*3^(1/4)*x^5*log(-(6*sqrt(3)*(2*x^4 - x - 1)^(1/4
)*x^3 + 6*I*3^(1/4)*sqrt(2*x^4 - x - 1)*x^2 - 3^(3/4)*(5*I*x^4 - I*x - I) - 6*(2*x^4 - x - 1)^(3/4)*x)/(x^4 +
x + 1)) + 5*I*3^(1/4)*x^5*log(-(6*sqrt(3)*(2*x^4 - x - 1)^(1/4)*x^3 - 6*I*3^(1/4)*sqrt(2*x^4 - x - 1)*x^2 - 3^
(3/4)*(-5*I*x^4 + I*x + I) - 6*(2*x^4 - x - 1)^(3/4)*x)/(x^4 + x + 1)) - 5*3^(1/4)*x^5*log((6*sqrt(3)*(2*x^4 -
x - 1)^(1/4)*x^3 - 6*3^(1/4)*sqrt(2*x^4 - x - 1)*x^2 - 3^(3/4)*(5*x^4 - x - 1) + 6*(2*x^4 - x - 1)^(3/4)*x)/(
x^4 + x + 1)) - 4*(12*x^4 - x - 1)*(2*x^4 - x - 1)^(1/4))/x^5

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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