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

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 = 0.53, size = 78, normalized size = 1.00 \begin {gather*} -\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 ) \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 = 14.72, 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 = 3.05, size = 1594, normalized size = 20.44

result too large to display

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)

[Out]

-4*(2*x^4-x-2)^(1/4)/x+(RootOf(_Z^4-3)*ln(-(20*RootOf(_Z^4-3)^2*x^12+8*(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)^(1/4)*RootOf(_Z^4-3)^3*x^9-24*RootOf(_Z^4-3)^2*x^9-8*(8*x^12-12*x^9-24*x^8+6*x^6+24*x^5+2
4*x^4-x^3-6*x^2-12*x-8)^(1/4)*RootOf(_Z^4-3)^3*x^6-48*x^8*RootOf(_Z^4-3)^2-16*(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)^(1/4)*RootOf(_Z^4-3)^3*x^5+12*(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)^(1/2)*x^6+9*RootOf(_Z^4-3)^2*x^6+6*(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)^(3/4)*R
ootOf(_Z^4-3)*x^3+2*(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)^(1/4)*RootOf(_Z^4-3)^3*x^3+36*
RootOf(_Z^4-3)^2*x^5+8*(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)^(1/4)*RootOf(_Z^4-3)^3*x^2+
36*RootOf(_Z^4-3)^2*x^4-6*(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)^(1/2)*x^3+8*(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)^(1/4)*RootOf(_Z^4-3)^3*x-x^3*RootOf(_Z^4-3)^2-12*(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)^(1/2)*x^2-6*x^2*RootOf(_Z^4-3)^2-12*RootOf(_Z^4-3)^2*x-8*RootOf
(_Z^4-3)^2)/(x^4+x+2)/(2*x^4-x-2)^2)+RootOf(_Z^2+RootOf(_Z^4-3)^2)*ln(-(-20*RootOf(_Z^4-3)^2*x^12-8*(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)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^9+24*
RootOf(_Z^4-3)^2*x^9+8*(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)^(1/4)*RootOf(_Z^2+RootOf(_Z
^4-3)^2)*RootOf(_Z^4-3)^2*x^6+48*x^8*RootOf(_Z^4-3)^2+16*(8*x^12-12*x^9-24*x^8+6*x^6+24*x^5+24*x^4-x^3-6*x^2-1
2*x-8)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^5+12*(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)^(1/2)*x^6-9*RootOf(_Z^4-3)^2*x^6+6*(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
)^(3/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*x^3-2*(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)^(1/4)*
RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^3-36*RootOf(_Z^4-3)^2*x^5-8*(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)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^2-36*RootOf(_Z^4-3)^2*x^4-6*
(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)^(1/2)*x^3-8*(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)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x+x^3*RootOf(_Z^4-3)^2-12*(8*x^12-1
2*x^9-24*x^8+6*x^6+24*x^5+24*x^4-x^3-6*x^2-12*x-8)^(1/2)*x^2+6*x^2*RootOf(_Z^4-3)^2+12*RootOf(_Z^4-3)^2*x+8*Ro
otOf(_Z^4-3)^2)/(x^4+x+2)/(2*x^4-x-2)^2))/(2*x^4-x-2)^(3/4)*((2*x^4-x-2)^3)^(1/4)

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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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