\(\int \frac {(1+2 x^8) \sqrt [4]{-1-2 x^4+2 x^8} (1-3 x^8+4 x^{16})}{x^{10} (-1+2 x^8)} \, dx\) [2279]

Optimal result

Integrand size = 48, antiderivative size = 173 \[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\frac {\sqrt [4]{-1-2 x^4+2 x^8} \left (5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right )}{45 x^9}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2} x^2-\sqrt {-1-2 x^4+2 x^8}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-2 x^4+2 x^8}}{2 x^2+\sqrt {2} \sqrt {-1-2 x^4+2 x^8}}\right )}{2 \sqrt [4]{2}} \]

[Out]

1/45*(2*x^8-2*x^4-1)^(1/4)*(20*x^16-4*x^12+9*x^8+2*x^4+5)/x^9-1/4*arctan(2^(3/4)*x*(2*x^8-2*x^4-1)^(1/4)/(2^(1
/2)*x^2-(2*x^8-2*x^4-1)^(1/2)))*2^(3/4)-1/4*arctanh(2*2^(1/4)*x*(2*x^8-2*x^4-1)^(1/4)/(2*x^2+2^(1/2)*(2*x^8-2*
x^4-1)^(1/2)))*2^(3/4)

Rubi [F]

\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx \]

[In]

Int[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]

[Out]

((-1 - 2*x^4 + 2*x^8)^(1/4)*AppellF1[-9/4, -1/4, -1/4, -5/4, (2*x^4)/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(9
*x^9*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)) + ((-1 - 2*x^4 + 2*x^8)^(1/4)*Appell
F1[-1/4, -1/4, -1/4, 3/4, (2*x^4)/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(x*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*
(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)) + (4*x^7*(-1 - 2*x^4 + 2*x^8)^(1/4)*AppellF1[7/4, -1/4, -1/4, 11/4, (2*x^4)
/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(7*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)
) + ((I/2)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(I - 2^(1/8)*x), x])/2^(3/4) - Defer[Int][(-1 - 2*x^4 + 2*x^8
)^(1/4)/(1 - 2^(1/8)*x), x]/(2*2^(3/4)) - ((1/4 - I/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/((-1)^(1/4) - 2^(
1/8)*x), x])/2^(1/4) - ((-1)^(1/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(-(-1)^(3/4) - 2^(1/8)*x), x])/(2*2^(
3/4)) + ((I/2)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(I + 2^(1/8)*x), x])/2^(3/4) - Defer[Int][(-1 - 2*x^4 + 2
*x^8)^(1/4)/(1 + 2^(1/8)*x), x]/(2*2^(3/4)) - ((1/4 - I/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/((-1)^(1/4) +
2^(1/8)*x), x])/2^(1/4) - ((-1)^(1/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(-(-1)^(3/4) + 2^(1/8)*x), x])/(2
*2^(3/4))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^{10}}-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^2}+4 x^6 \sqrt [4]{-1-2 x^4+2 x^8}+\frac {4 x^6 \sqrt [4]{-1-2 x^4+2 x^8}}{-1+2 x^8}\right ) \, dx \\ & = 4 \int x^6 \sqrt [4]{-1-2 x^4+2 x^8} \, dx+4 \int \frac {x^6 \sqrt [4]{-1-2 x^4+2 x^8}}{-1+2 x^8} \, dx-\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^{10}} \, dx-\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^2} \, dx \\ & = 4 \int \left (\frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-\sqrt {2}+2 x^4\right )}+\frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt {2}+2 x^4\right )}\right ) \, dx-\frac {\sqrt [4]{-1-2 x^4+2 x^8} \int \frac {\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}{x^{10}} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}-\frac {\sqrt [4]{-1-2 x^4+2 x^8} \int \frac {\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}+\frac {\left (4 \sqrt [4]{-1-2 x^4+2 x^8}\right ) \int x^6 \sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}} \\ & = \frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {9}{4},-\frac {1}{4},-\frac {1}{4},-\frac {5}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (\frac {7}{4},-\frac {1}{4},-\frac {1}{4},\frac {11}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+2 \int \frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{-\sqrt {2}+2 x^4} \, dx+2 \int \frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2}+2 x^4} \, dx \\ & = \frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {9}{4},-\frac {1}{4},-\frac {1}{4},-\frac {5}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (\frac {7}{4},-\frac {1}{4},-\frac {1}{4},\frac {11}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+2 \int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (i-\sqrt [4]{2} x^2\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (i+\sqrt [4]{2} x^2\right )}\right ) \, dx+2 \int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (1-\sqrt [4]{2} x^2\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (1+\sqrt [4]{2} x^2\right )}\right ) \, dx \\ & = \frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {9}{4},-\frac {1}{4},-\frac {1}{4},-\frac {5}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (\frac {7}{4},-\frac {1}{4},-\frac {1}{4},\frac {11}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i-\sqrt [4]{2} x^2} \, dx}{2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1-\sqrt [4]{2} x^2} \, dx}{2^{3/4}}+\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i+\sqrt [4]{2} x^2} \, dx}{2^{3/4}}+\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1+\sqrt [4]{2} x^2} \, dx}{2^{3/4}} \\ & = \frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {9}{4},-\frac {1}{4},-\frac {1}{4},-\frac {5}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (\frac {7}{4},-\frac {1}{4},-\frac {1}{4},\frac {11}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\int \left (\frac {i \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (i-\sqrt [8]{2} x\right )}+\frac {i \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (i+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}-\frac {\int \left (\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (1-\sqrt [8]{2} x\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (1+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}-\frac {\int \left (-\frac {(-1)^{3/4} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt [4]{-1}-\sqrt [8]{2} x\right )}-\frac {(-1)^{3/4} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt [4]{-1}+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}+\frac {\int \left (-\frac {\sqrt [4]{-1} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-(-1)^{3/4}-\sqrt [8]{2} x\right )}-\frac {\sqrt [4]{-1} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-(-1)^{3/4}+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}} \\ & = \frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {9}{4},-\frac {1}{4},-\frac {1}{4},-\frac {5}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} \operatorname {AppellF1}\left (\frac {7}{4},-\frac {1}{4},-\frac {1}{4},\frac {11}{4},\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {i \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}+\frac {i \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\sqrt [4]{-1} \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{-(-1)^{3/4}-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\sqrt [4]{-1} \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{-(-1)^{3/4}+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}+-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt [4]{-1}-\sqrt [8]{2} x} \, dx}{\sqrt [4]{2}}+-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt [4]{-1}+\sqrt [8]{2} x} \, dx}{\sqrt [4]{2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.68 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\frac {1}{180} \left (\frac {4 \sqrt [4]{-1-2 x^4+2 x^8} \left (5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right )}{x^9}-45\ 2^{3/4} \arctan \left (\frac {2^{3/4} x \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2} x^2-\sqrt {-1-2 x^4+2 x^8}}\right )-45\ 2^{3/4} \text {arctanh}\left (\frac {2 x \sqrt [4]{-2-4 x^4+4 x^8}}{2 x^2+\sqrt {-2-4 x^4+4 x^8}}\right )\right ) \]

[In]

Integrate[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]

[Out]

((4*(-1 - 2*x^4 + 2*x^8)^(1/4)*(5 + 2*x^4 + 9*x^8 - 4*x^12 + 20*x^16))/x^9 - 45*2^(3/4)*ArcTan[(2^(3/4)*x*(-1
- 2*x^4 + 2*x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - 2*x^4 + 2*x^8])] - 45*2^(3/4)*ArcTanh[(2*x*(-2 - 4*x^4 + 4*x^
8)^(1/4))/(2*x^2 + Sqrt[-2 - 4*x^4 + 4*x^8])])/180

Maple [A] (verified)

Time = 150.03 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {-45 x^{9} \left (2 \arctan \left (\frac {2^{\frac {1}{4}} \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )+\ln \left (\frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {2 x^{8}-2 x^{4}-1}}{-\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {2 x^{8}-2 x^{4}-1}}\right )\right ) 2^{\frac {3}{4}}+8 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{360 x^{9}}\)	\(197\)
trager	\(\frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{45 x^{9}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \sqrt {2 x^{8}-2 x^{4}-1}\, x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )}{2 x^{8}-1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \sqrt {2 x^{8}-2 x^{4}-1}\, x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3}}{2 x^{8}-1}\right )}{4}\)	\(348\)
risch	\(\text {Expression too large to display}\)	\(1128\)

[In]

int((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x,method=_RETURNVERBOSE)

[Out]

1/360*(-45*x^9*(2*arctan((2^(1/4)*(2*x^8-2*x^4-1)^(1/4)-x)/x)+2*arctan((2^(1/4)*(2*x^8-2*x^4-1)^(1/4)+x)/x)+ln
(((2*x^8-2*x^4-1)^(1/4)*2^(3/4)*x+2^(1/2)*x^2+(2*x^8-2*x^4-1)^(1/2))/(-(2*x^8-2*x^4-1)^(1/4)*2^(3/4)*x+2^(1/2)
*x^2+(2*x^8-2*x^4-1)^(1/2))))*2^(3/4)+8*(2*x^8-2*x^4-1)^(1/4)*(20*x^16-4*x^12+9*x^8+2*x^4+5))/x^9

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 154.25 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.71 \[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\frac {-\left (45 i + 45\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{8} + \left (4 i - 4\right ) \, x^{4} + i - 1\right )} + 8 \, {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{2 \, x^{8} - 1}\right ) + \left (45 i - 45\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {-\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{8} - \left (4 i + 4\right ) \, x^{4} - i - 1\right )} + 8 \, {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{2 \, x^{8} - 1}\right ) - \left (45 i - 45\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{8} + \left (4 i + 4\right ) \, x^{4} + i + 1\right )} + 8 \, {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{2 \, x^{8} - 1}\right ) + \left (45 i + 45\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {-\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{8} - \left (4 i - 4\right ) \, x^{4} - i + 1\right )} + 8 \, {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{2 \, x^{8} - 1}\right ) + 64 \, {\left (20 \, x^{16} - 4 \, x^{12} + 9 \, x^{8} + 2 \, x^{4} + 5\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{2880 \, x^{9}} \]

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="fricas")

[Out]

1/2880*(-(45*I + 45)*8^(3/4)*sqrt(2)*x^9*log(((I + 1)*8^(3/4)*sqrt(2)*sqrt(2*x^8 - 2*x^4 - 1)*x^2 + 8*I*sqrt(2
)*(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 8^(1/4)*sqrt(2)*(-(2*I - 2)*x^8 + (4*I - 4)*x^4 + I - 1) + 8*(2*x^8 - 2*x^4
- 1)^(3/4)*x)/(2*x^8 - 1)) + (45*I - 45)*8^(3/4)*sqrt(2)*x^9*log((-(I - 1)*8^(3/4)*sqrt(2)*sqrt(2*x^8 - 2*x^4
- 1)*x^2 - 8*I*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 8^(1/4)*sqrt(2)*((2*I + 2)*x^8 - (4*I + 4)*x^4 - I - 1)
+ 8*(2*x^8 - 2*x^4 - 1)^(3/4)*x)/(2*x^8 - 1)) - (45*I - 45)*8^(3/4)*sqrt(2)*x^9*log(((I - 1)*8^(3/4)*sqrt(2)*
sqrt(2*x^8 - 2*x^4 - 1)*x^2 - 8*I*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 8^(1/4)*sqrt(2)*(-(2*I + 2)*x^8 + (4
*I + 4)*x^4 + I + 1) + 8*(2*x^8 - 2*x^4 - 1)^(3/4)*x)/(2*x^8 - 1)) + (45*I + 45)*8^(3/4)*sqrt(2)*x^9*log((-(I
+ 1)*8^(3/4)*sqrt(2)*sqrt(2*x^8 - 2*x^4 - 1)*x^2 + 8*I*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 8^(1/4)*sqrt(2)
*((2*I - 2)*x^8 - (4*I - 4)*x^4 - I + 1) + 8*(2*x^8 - 2*x^4 - 1)^(3/4)*x)/(2*x^8 - 1)) + 64*(20*x^16 - 4*x^12
+ 9*x^8 + 2*x^4 + 5)*(2*x^8 - 2*x^4 - 1)^(1/4))/x^9

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((2*x**8+1)*(2*x**8-2*x**4-1)**(1/4)*(4*x**16-3*x**8+1)/x**10/(2*x**8-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{16} - 3 \, x^{8} + 1\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (2 \, x^{8} + 1\right )}}{{\left (2 \, x^{8} - 1\right )} x^{10}} \,d x } \]

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="maxima")

[Out]

integrate((4*x^16 - 3*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(2*x^8 + 1)/((2*x^8 - 1)*x^10), x)

Giac [F]

\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{16} - 3 \, x^{8} + 1\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (2 \, x^{8} + 1\right )}}{{\left (2 \, x^{8} - 1\right )} x^{10}} \,d x } \]

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="giac")

[Out]

integrate((4*x^16 - 3*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(2*x^8 + 1)/((2*x^8 - 1)*x^10), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx=\int \frac {\left (2\,x^8+1\right )\,{\left (2\,x^8-2\,x^4-1\right )}^{1/4}\,\left (4\,x^{16}-3\,x^8+1\right )}{x^{10}\,\left (2\,x^8-1\right )} \,d x \]

[In]

int(((2*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(4*x^16 - 3*x^8 + 1))/(x^10*(2*x^8 - 1)),x)

[Out]

int(((2*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(4*x^16 - 3*x^8 + 1))/(x^10*(2*x^8 - 1)), x)