\(\int \frac {(4+x^3) (1+2 x^3+x^6+x^8)}{x^4 \sqrt [4]{1+x^3} (-1-2 x^3-x^6+x^8)} \, dx\) [1844]

Optimal result

Integrand size = 48, antiderivative size = 125 \[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-2 \arctan \left (\frac {x}{\sqrt [4]{1+x^3}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{-x^2+\sqrt {1+x^3}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{1+x^3}}{x}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 11.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+2 \left (\arctan \left (\frac {\sqrt [4]{1+x^3}}{x}\right )-\frac {\arctan \left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}}\right ) \]

[In]

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

[Out]

(4*(1 + x^3)^(3/4))/(3*x^3) + 2*(ArcTan[(1 + x^3)^(1/4)/x] - ArcTan[(-x^2 + Sqrt[1 + x^3])/(Sqrt[2]*x*(1 + x^3
)^(1/4))]/Sqrt[2] - ArcTanh[x/(1 + x^3)^(1/4)] + ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]/Sq
rt[2])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 11.32 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.02

method	result	size
trager	\(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )+\ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}-1}{x^{4}-x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{3}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}-x^{3}-1}\right )\)	\(377\)
risch	\(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}-1}\right )+\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}-1}{x^{4}-x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )\)	\(432\)

[In]

int((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

4/3*(x^3+1)^(3/4)/x^3+RootOf(_Z^4+1)^3*ln((-RootOf(_Z^4+1)^3*x^4+2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3+RootOf(_
Z^4+1)^3*x^3-2*(x^3+1)^(1/2)*RootOf(_Z^4+1)*x^2+2*(x^3+1)^(3/4)*x+RootOf(_Z^4+1)^3)/(x^4+x^3+1))+RootOf(_Z^4+1
)*ln((-2*(x^3+1)^(1/2)*RootOf(_Z^4+1)^3*x^2-2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)*x^4+2*(x^3+1)^
(3/4)*x+RootOf(_Z^4+1)*x^3+RootOf(_Z^4+1))/(x^4+x^3+1))+ln((2*(x^3+1)^(3/4)*x-2*x^2*(x^3+1)^(1/2)+2*(x^3+1)^(1
/4)*x^3-x^4-x^3-1)/(x^4-x^3-1))+RootOf(_Z^4+1)^2*ln(-(2*RootOf(_Z^4+1)^2*(x^3+1)^(1/2)*x^2-RootOf(_Z^4+1)^2*x^
4-RootOf(_Z^4+1)^2*x^3+2*(x^3+1)^(3/4)*x-2*(x^3+1)^(1/4)*x^3-RootOf(_Z^4+1)^2)/(x^4-x^3-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 28.74 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.30 \[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\frac {-\left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} - 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} + i + 1\right )}}{x^{4} + x^{3} + 1}\right ) + \left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} - 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} - i - 1\right )}}{x^{4} + x^{3} + 1}\right ) + \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} - 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} - i + 1\right )}}{x^{4} + x^{3} + 1}\right ) - \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} - 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} + i - 1\right )}}{x^{4} + x^{3} + 1}\right ) + 12 \, x^{3} \arctan \left (\frac {2 \, {\left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} - 1}\right ) + 12 \, x^{3} \log \left (\frac {x^{4} - 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} + 1} x^{2} - 2 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} - x^{3} - 1}\right ) + 16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \]

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="fricas")

[Out]

1/12*(-(3*I + 3)*sqrt(2)*x^3*log((4*I*(x^3 + 1)^(1/4)*x^3 - (2*I - 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 - 4*(x^3 + 1)^
(3/4)*x + sqrt(2)*(-(I + 1)*x^4 + (I + 1)*x^3 + I + 1))/(x^4 + x^3 + 1)) + (3*I + 3)*sqrt(2)*x^3*log((4*I*(x^3
+ 1)^(1/4)*x^3 + (2*I - 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 - 4*(x^3 + 1)^(3/4)*x + sqrt(2)*((I + 1)*x^4 - (I + 1)*x
^3 - I - 1))/(x^4 + x^3 + 1)) + (3*I - 3)*sqrt(2)*x^3*log((-4*I*(x^3 + 1)^(1/4)*x^3 + (2*I + 2)*sqrt(2)*sqrt(x
^3 + 1)*x^2 - 4*(x^3 + 1)^(3/4)*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 - I + 1))/(x^4 + x^3 + 1)) - (3*I - 3)*
sqrt(2)*x^3*log((-4*I*(x^3 + 1)^(1/4)*x^3 - (2*I + 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 - 4*(x^3 + 1)^(3/4)*x + sqrt(2
)*(-(I - 1)*x^4 + (I - 1)*x^3 + I - 1))/(x^4 + x^3 + 1)) + 12*x^3*arctan(2*((x^3 + 1)^(1/4)*x^3 + (x^3 + 1)^(3
/4)*x)/(x^4 - x^3 - 1)) + 12*x^3*log((x^4 - 2*(x^3 + 1)^(1/4)*x^3 + x^3 + 2*sqrt(x^3 + 1)*x^2 - 2*(x^3 + 1)^(3
/4)*x + 1)/(x^4 - x^3 - 1)) + 16*(x^3 + 1)^(3/4))/x^3

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**3+4)*(x**8+x**6+2*x**3+1)/x**4/(x**3+1)**(1/4)/(x**8-x**6-2*x**3-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 4)/((x^8 - x^6 - 2*x^3 - 1)*(x^3 + 1)^(1/4)*x^4), x)

Giac [F]

\[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]

[In]

integrate((x^3+4)*(x^8+x^6+2*x^3+1)/x^4/(x^3+1)^(1/4)/(x^8-x^6-2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^8 + x^6 + 2*x^3 + 1)*(x^3 + 4)/((x^8 - x^6 - 2*x^3 - 1)*(x^3 + 1)^(1/4)*x^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\int -\frac {\left (x^3+4\right )\,\left (x^8+x^6+2\,x^3+1\right )}{x^4\,{\left (x^3+1\right )}^{1/4}\,\left (-x^8+x^6+2\,x^3+1\right )} \,d x \]

[In]

int(-((x^3 + 4)*(2*x^3 + x^6 + x^8 + 1))/(x^4*(x^3 + 1)^(1/4)*(2*x^3 + x^6 - x^8 + 1)),x)

[Out]

int(-((x^3 + 4)*(2*x^3 + x^6 + x^8 + 1))/(x^4*(x^3 + 1)^(1/4)*(2*x^3 + x^6 - x^8 + 1)), x)