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

Optimal. Leaf size=173 \[ -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{2 x^8-2 x^4-1}}{\sqrt {2} x^2-\sqrt {2 x^8-2 x^4-1}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{2 x^8-2 x^4-1}}{2 x^2+\sqrt {2} \sqrt {2 x^8-2 x^4-1}}\right )}{2 \sqrt [4]{2}}+\frac {\sqrt [4]{2 x^8-2 x^4-1} \left (20 x^{16}-4 x^{12}+9 x^8+2 x^4+5\right )}{45 x^9} \]

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Rubi [F]  time = 3.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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*}

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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]

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]

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IntegrateAlgebraic [A]  time = 2.88, size = 173, normalized size = 1.00 \begin {gather*} \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 {\tan ^{-1}\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 {\tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 620.10, size = 347, normalized size = 2.01

method	result	size
trager	\(\frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{45 x^{9}}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{8}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {2 x^{8}-2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{2 x^{8}-1}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{8}+4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {2 x^{8}-2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{2 x^{8}-1}\right )}{4}\)	\(347\)
risch	\(\frac {40 x^{24}-48 x^{20}+6 x^{16}-10 x^{12}-3 x^{8}-12 x^{4}-5}{45 x^{9} \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {-8 x^{24} \RootOf \left (\textit {\_Z}^{4}+2\right )+32 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{20}-8 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{17}-28 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{16}+16 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{13}-4 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{10}-16 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{12}+4 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{6}+14 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{8}-8 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}+2 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}+4 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {3}{4}} x^{3}+8 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{4}-2 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x +\RootOf \left (\textit {\_Z}^{4}+2\right )}{\left (2 x^{8}-1\right ) \left (2 x^{8}-2 x^{4}-1\right )^{2}}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{24}+32 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{20}+8 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{17}-28 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{16}-16 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{13}+4 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{10}-16 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{12}-4 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+14 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{8}+8 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-2 \sqrt {8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+4 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {3}{4}} x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}+2 \left (8 x^{24}-24 x^{20}+12 x^{16}+16 x^{12}-6 x^{8}-6 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (2 x^{8}-1\right ) \left (2 x^{8}-2 x^{4}-1\right )^{2}}\right )}{4}\right ) \left (\left (2 x^{8}-2 x^{4}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}}}\)	\(1126\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/45*(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*RootOf(_Z^4+2)*ln(-(2*RootOf(_Z^4+2)^3*x^8-4
*RootOf(_Z^4+2)^3*x^4+4*(2*x^8-2*x^4-1)^(1/4)*RootOf(_Z^4+2)^2*x^3-4*(2*x^8-2*x^4-1)^(1/2)*RootOf(_Z^4+2)*x^2+
4*(2*x^8-2*x^4-1)^(3/4)*x-RootOf(_Z^4+2)^3)/(2*x^8-1))+1/4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(-2*RootOf(_Z^4+2
)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^8+4*RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^4-4*(2*x^8-2*x^4-1)^(
1/4)*RootOf(_Z^4+2)^2*x^3-4*(2*x^8-2*x^4-1)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+4*(2*x^8-2*x^4-1)^(3/4)*x+
RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2))/(2*x^8-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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