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

Optimal. Leaf size=78 \[ -\frac {4 \sqrt [4]{2 x^4-x-2}}{x}-2 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-2}}\right )+2 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-2}}\right ) \]

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Rubi [F]  time = 1.07, 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 {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.52, size = 78, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 14.98, size = 287, normalized size = 3.68 \begin {gather*} \frac {4 \cdot 3^{\frac {1}{4}} x \arctan \left (\frac {6 \cdot 3^{\frac {3}{4}} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x + 3^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} + 3^{\frac {1}{4}} {\left (5 \, x^{4} - x - 2\right )}\right )}}{3 \, {\left (x^{4} + x + 2\right )}}\right ) + 3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} + 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} - 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 8 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*3^(1/4)*x*arctan(1/3*(6*3^(3/4)*(2*x^4 - x - 2)^(1/4)*x^3 + 6*3^(1/4)*(2*x^4 - x - 2)^(3/4)*x + 3^(3/4)
*(2*3^(3/4)*sqrt(2*x^4 - x - 2)*x^2 + 3^(1/4)*(5*x^4 - x - 2)))/(x^4 + x + 2)) + 3^(1/4)*x*log((6*sqrt(3)*(2*x
^4 - x - 2)^(1/4)*x^3 + 6*3^(1/4)*sqrt(2*x^4 - x - 2)*x^2 + 3^(3/4)*(5*x^4 - x - 2) + 6*(2*x^4 - x - 2)^(3/4)*
x)/(x^4 + x + 2)) - 3^(1/4)*x*log((6*sqrt(3)*(2*x^4 - x - 2)^(1/4)*x^3 - 6*3^(1/4)*sqrt(2*x^4 - x - 2)*x^2 - 3
^(3/4)*(5*x^4 - x - 2) + 6*(2*x^4 - x - 2)^(3/4)*x)/(x^4 + x + 2)) - 8*(2*x^4 - x - 2)^(1/4))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} {\left (3 \, x + 8\right )}}{{\left (x^{4} + x + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 9.74, size = 303, normalized size = 3.88

method	result	size
trager	\(-\frac {4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}-\RootOf \left (\textit {\_Z}^{4}-3\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{4}+6 \left (2 x^{4}-x -2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {2 x^{4}-x -2}\, \RootOf \left (\textit {\_Z}^{4}-3\right ) x^{2}+6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x +2 \RootOf \left (\textit {\_Z}^{4}-3\right )^{3}}{x^{4}+x +2}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{4}-6 \left (2 x^{4}-x -2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {2 x^{4}-x -2}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{2}+6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right )}{x^{4}+x +2}\right )\)	\(303\)
risch	\(-\frac {4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}+\frac {\left (-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \ln \left (-\frac {-20 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{12}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{9}+24 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{9}-8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{6}+48 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{8}-16 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{5}-9 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{6}+12 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{6}+2 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{3}-36 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{5}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{2}-36 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{4}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x +\RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{3}+6 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{2}-12 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{2}+12 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x +8 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2}}{\left (x^{4}+x +2\right ) \left (2 x^{4}-x -2\right )^{2}}\right )-\RootOf \left (\textit {\_Z}^{4}-3\right ) \ln \left (\frac {-20 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{12}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{9}+24 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{9}-8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{6}+48 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{8}-16 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{5}-9 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{6}-12 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{6}+2 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{3}-36 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{5}+6 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right ) x^{3}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{2}-36 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{4}+8 \left (8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}+6 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{3}+6 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{2}+12 \sqrt {8 x^{12}-12 x^{9}-24 x^{8}+6 x^{6}+24 x^{5}+24 x^{4}-x^{3}-6 x^{2}-12 x -8}\, x^{2}+12 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x +8 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2}}{\left (x^{4}+x +2\right ) \left (2 x^{4}-x -2\right )^{2}}\right )\right ) \left (\left (2 x^{4}-x -2\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{4}-x -2\right )^{\frac {3}{4}}}\)	\(1594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(2*x^4-x-2)^(1/4)/x-RootOf(_Z^4-3)*ln((-5*RootOf(_Z^4-3)^3*x^4+6*(2*x^4-x-2)^(1/4)*RootOf(_Z^4-3)^2*x^3-6*(
2*x^4-x-2)^(1/2)*RootOf(_Z^4-3)*x^2+6*(2*x^4-x-2)^(3/4)*x+RootOf(_Z^4-3)^3*x+2*RootOf(_Z^4-3)^3)/(x^4+x+2))-Ro
otOf(_Z^2+RootOf(_Z^4-3)^2)*ln((5*RootOf(_Z^4-3)^2*RootOf(_Z^2+RootOf(_Z^4-3)^2)*x^4-6*(2*x^4-x-2)^(1/4)*RootO
f(_Z^4-3)^2*x^3-6*(2*x^4-x-2)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*x^2+6*(2*x^4-x-2)^(3/4)*x-RootOf(_Z^4-3)^2*R
ootOf(_Z^2+RootOf(_Z^4-3)^2)*x-2*RootOf(_Z^4-3)^2*RootOf(_Z^2+RootOf(_Z^4-3)^2))/(x^4+x+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} {\left (3 \, x + 8\right )}}{{\left (x^{4} + x + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (3\,x+8\right )\,{\left (2\,x^4-x-2\right )}^{1/4}}{x^2\,\left (x^4+x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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