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

Optimal. Leaf size=90 \[ -4 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-1}}\right )+4 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-1}}\right )-\frac {4 \sqrt [4]{2 x^4-x-1} \left (12 x^4-x-1\right )}{5 x^5} \]

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Rubi [F]  time = 1.71, 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 {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-4*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x^6, x] - 3*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x^5, x] + 8*Defer[Int][(-1
- x + 2*x^4)^(1/4)/x^2, x] - 2*Defer[Int][(-1 - x + 2*x^4)^(1/4)/x, x] + 2*Defer[Int][(-1 - x + 2*x^4)^(1/4)/(
1 + x + x^4), x] - 8*Defer[Int][(x^2*(-1 - x + 2*x^4)^(1/4))/(1 + x + x^4), x] + 2*Defer[Int][(x^3*(-1 - x + 2
*x^4)^(1/4))/(1 + x + x^4), x]

Rubi steps

\begin {align*} \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx &=\int \left (-\frac {4 \sqrt [4]{-1-x+2 x^4}}{x^6}-\frac {3 \sqrt [4]{-1-x+2 x^4}}{x^5}+\frac {8 \sqrt [4]{-1-x+2 x^4}}{x^2}-\frac {2 \sqrt [4]{-1-x+2 x^4}}{x}+\frac {2 \left (1-4 x^2+x^3\right ) \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \frac {\left (1-4 x^2+x^3\right ) \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \left (\frac {\sqrt [4]{-1-x+2 x^4}}{1+x+x^4}-\frac {4 x^2 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}+\frac {x^3 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4}\right ) \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x} \, dx\right )+2 \int \frac {\sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx+2 \int \frac {x^3 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx-3 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^5} \, dx-4 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^6} \, dx+8 \int \frac {\sqrt [4]{-1-x+2 x^4}}{x^2} \, dx-8 \int \frac {x^2 \sqrt [4]{-1-x+2 x^4}}{1+x+x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.42, size = 90, normalized size = 1.00 \begin {gather*} -4 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-1}}\right )+4 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2 x^4-x-1}}\right )-\frac {4 \sqrt [4]{2 x^4-x-1} \left (12 x^4-x-1\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 - x + 2*x^4)^(1/4)*(-1 - x + 12*x^4))/(5*x^5) - 4*3^(1/4)*ArcTan[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)] +
4*3^(1/4)*ArcTanh[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)]

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fricas [B]  time = 13.54, size = 304, normalized size = 3.38 \begin {gather*} \frac {20 \cdot 3^{\frac {1}{4}} x^{5} \arctan \left (\frac {6 \cdot 3^{\frac {3}{4}} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x + 3^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} + 3^{\frac {1}{4}} {\left (5 \, x^{4} - x - 1\right )}\right )}}{3 \, {\left (x^{4} + x + 1\right )}}\right ) + 5 \cdot 3^{\frac {1}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} + 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 1\right )} + 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) - 5 \cdot 3^{\frac {1}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 1} x^{2} - 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 1\right )} + 6 \, {\left (2 \, x^{4} - x - 1\right )}^{\frac {3}{4}} x}{x^{4} + x + 1}\right ) - 4 \, {\left (12 \, x^{4} - x - 1\right )} {\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(20*3^(1/4)*x^5*arctan(1/3*(6*3^(3/4)*(2*x^4 - x - 1)^(1/4)*x^3 + 6*3^(1/4)*(2*x^4 - x - 1)^(3/4)*x + 3^(3
/4)*(2*3^(3/4)*sqrt(2*x^4 - x - 1)*x^2 + 3^(1/4)*(5*x^4 - x - 1)))/(x^4 + x + 1)) + 5*3^(1/4)*x^5*log((6*sqrt(
3)*(2*x^4 - x - 1)^(1/4)*x^3 + 6*3^(1/4)*sqrt(2*x^4 - x - 1)*x^2 + 3^(3/4)*(5*x^4 - x - 1) + 6*(2*x^4 - x - 1)
^(3/4)*x)/(x^4 + x + 1)) - 5*3^(1/4)*x^5*log((6*sqrt(3)*(2*x^4 - x - 1)^(1/4)*x^3 - 6*3^(1/4)*sqrt(2*x^4 - x -
1)*x^2 - 3^(3/4)*(5*x^4 - x - 1) + 6*(2*x^4 - x - 1)^(3/4)*x)/(x^4 + x + 1)) - 4*(12*x^4 - x - 1)*(2*x^4 - x
- 1)^(1/4))/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} - x - 1\right )}^{\frac {1}{4}} {\left (x^{4} - x - 1\right )} {\left (3 \, x + 4\right )}}{{\left (x^{4} + x + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.14, size = 1633, normalized size = 18.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*(24*x^8-14*x^5-14*x^4+x^2+2*x+1)/x^5/(2*x^4-x-1)^(3/4)+(-2*RootOf(_Z^4-3)*ln((-20*RootOf(_Z^4-3)^2*x^12+8
*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^4-3)^3*x^9+24*RootOf(_Z^4-3)^2*x^9-
8*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^4-3)^3*x^6+24*x^8*RootOf(_Z^4-3)^2
-8*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^4-3)^3*x^5-9*RootOf(_Z^4-3)^2*x^6
-12*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*x^6+2*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6
*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^4-3)^3*x^3-18*RootOf(_Z^4-3)^2*x^5+6*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+
6*x^4-x^3-3*x^2-3*x-1)^(3/4)*RootOf(_Z^4-3)*x^3+4*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1
/4)*RootOf(_Z^4-3)^3*x^2-9*RootOf(_Z^4-3)^2*x^4+2*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1
/4)*RootOf(_Z^4-3)^3*x+x^3*RootOf(_Z^4-3)^2+6*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*
x^3+3*x^2*RootOf(_Z^4-3)^2+6*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*x^2+3*RootOf(_Z^4
-3)^2*x+RootOf(_Z^4-3)^2)/(2*x^3+2*x^2+2*x+1)^2/(x^4+x+1)/(-1+x)^2)-2*RootOf(_Z^2+RootOf(_Z^4-3)^2)*ln(-(-20*R
ootOf(_Z^4-3)^2*x^12+8*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(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^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1
)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^6+24*x^8*RootOf(_Z^4-3)^2-8*(8*x^12-12*x^9-12*x^8+6*x
^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^5-9*RootOf(_Z^4-3)^2*x
^6+12*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*x^6+2*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5
+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^3-6*(8*x^12-12*x^9-12*x^8+6*x^6
+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(3/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*x^3-18*RootOf(_Z^4-3)^2*x^5+4*(8*x^12-12*x^
9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)*RootOf(_Z^4-3)^2*x^2-9*RootOf
(_Z^4-3)^2*x^4+2*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-3)^2)
*RootOf(_Z^4-3)^2*x+x^3*RootOf(_Z^4-3)^2-6*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*x^3
+3*x^2*RootOf(_Z^4-3)^2-6*(8*x^12-12*x^9-12*x^8+6*x^6+12*x^5+6*x^4-x^3-3*x^2-3*x-1)^(1/2)*x^2+3*RootOf(_Z^4-3)
^2*x+RootOf(_Z^4-3)^2)/(2*x^3+2*x^2+2*x+1)^2/(x^4+x+1)/(-1+x)^2))/(2*x^4-x-1)^(3/4)*((2*x^4-x-1)^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 - 1\right )}^{\frac {1}{4}} {\left (x^{4} - x - 1\right )} {\left (3 \, x + 4\right )}}{{\left (x^{4} + x + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (3\,x+4\right )\,\left (-x^4+x+1\right )\,{\left (2\,x^4-x-1\right )}^{1/4}}{x^6\,\left (x^4+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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