3.15.2 \(\int \frac {(-1+3 x^4) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 (4+x+4 x^4)} \, dx\)

Optimal. Leaf size=100 \[ \frac {\sqrt {x^8+x^5+2 x^4+x+1}}{4 x}-\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {x^8+x^5+2 x^4+x+1}}{x^4+x+1}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {x^8+x^5+2 x^4+x+1}}{x^4+x+1}\right ) \]

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Rubi [F]  time = 1.45, 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+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*Defer[Int][Sqrt[1 + x + 2*x^4 + x^5 + x^8]/x^2, x] + Defer[Int][Sqrt[1 + x + 2*x^4 + x^5 + x^8]/x, x]/16
+ Defer[Int][Sqrt[1 + x + 2*x^4 + x^5 + x^8]/(-4 - x - 4*x^4), x]/16 + 4*Defer[Int][(x^2*Sqrt[1 + x + 2*x^4 +
x^5 + x^8])/(4 + x + 4*x^4), x] - Defer[Int][(x^3*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(4 + x + 4*x^4), x]/4

Rubi steps

\begin {align*} \int \frac {\left (-1+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx &=\int \left (-\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{4 x^2}+\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{16 x}+\frac {\left (-1+64 x^2-4 x^3\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{16 \left (4+x+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \frac {\left (-1+64 x^2-4 x^3\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \left (\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{-4-x-4 x^4}+\frac {64 x^2 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4}-\frac {4 x^3 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4}\right ) \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{-4-x-4 x^4} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx-\frac {1}{4} \int \frac {x^3 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx+4 \int \frac {x^2 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + 3*x^4)*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(x^2*(4 + x + 4*x^4)),x]

[Out]

Integrate[((-1 + 3*x^4)*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(x^2*(4 + x + 4*x^4)), x]

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IntegrateAlgebraic [A]  time = 0.36, size = 100, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{4 x}-\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + 3*x^4)*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(x^2*(4 + x + 4*x^4)),x]

[Out]

Sqrt[1 + x + 2*x^4 + x^5 + x^8]/(4*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(1 + x + x^4
)])/8 + ArcTanh[Sqrt[1 + x + 2*x^4 + x^5 + x^8]/(1 + x + x^4)]/8

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fricas [A]  time = 3.36, size = 94, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {3} x \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{4} - x + 2\right )}}{6 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1}}\right ) + x \log \left (\frac {2 \, x^{4} + x - 2 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} + 2}{x}\right ) - 4 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1}}{16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1)*(x^8+x^5+2*x^4+x+1)^(1/2)/x^2/(4*x^4+x+4),x, algorithm="fricas")

[Out]

-1/16*(sqrt(3)*x*arctan(1/6*sqrt(3)*(2*x^4 - x + 2)/sqrt(x^8 + x^5 + 2*x^4 + x + 1)) + x*log((2*x^4 + x - 2*sq
rt(x^8 + x^5 + 2*x^4 + x + 1) + 2)/x) - 4*sqrt(x^8 + x^5 + 2*x^4 + x + 1))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (4 \, x^{4} + x + 4\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1)*(x^8+x^5+2*x^4+x+1)^(1/2)/x^2/(4*x^4+x+4),x, algorithm="giac")

[Out]

integrate(sqrt(x^8 + x^5 + 2*x^4 + x + 1)*(3*x^4 - 1)/((4*x^4 + x + 4)*x^2), x)

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maple [C]  time = 1.59, size = 124, normalized size = 1.24

method	result	size
trager	\(\frac {\sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x}+\frac {\ln \left (-\frac {2 x^{4}+2 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}+x +2}{x}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+3\right )-6 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x^{4}+x +4}\right )}{16}\)	\(124\)
risch	\(\frac {\sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x}-\frac {\ln \left (-\frac {-2 x^{4}+2 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}-x -2}{x}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+3\right )+6 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x^{4}+x +4}\right )}{16}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4-1)*(x^8+x^5+2*x^4+x+1)^(1/2)/x^2/(4*x^4+x+4),x,method=_RETURNVERBOSE)

[Out]

1/4*(x^8+x^5+2*x^4+x+1)^(1/2)/x+1/16*ln(-(2*x^4+2*(x^8+x^5+2*x^4+x+1)^(1/2)+x+2)/x)-1/16*RootOf(_Z^2+3)*ln((2*
RootOf(_Z^2+3)*x^4-RootOf(_Z^2+3)*x+2*RootOf(_Z^2+3)-6*(x^8+x^5+2*x^4+x+1)^(1/2))/(4*x^4+x+4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (4 \, x^{4} + x + 4\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-1)*(x^8+x^5+2*x^4+x+1)^(1/2)/x^2/(4*x^4+x+4),x, algorithm="maxima")

[Out]

integrate(sqrt(x^8 + x^5 + 2*x^4 + x + 1)*(3*x^4 - 1)/((4*x^4 + x + 4)*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^4-1\right )\,\sqrt {x^8+x^5+2\,x^4+x+1}}{x^2\,\left (4\,x^4+x+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4 - 1)*(x + 2*x^4 + x^5 + x^8 + 1)^(1/2))/(x^2*(x + 4*x^4 + 4)),x)

[Out]

int(((3*x^4 - 1)*(x + 2*x^4 + x^5 + x^8 + 1)^(1/2))/(x^2*(x + 4*x^4 + 4)), 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((3*x**4-1)*(x**8+x**5+2*x**4+x+1)**(1/2)/x**2/(4*x**4+x+4),x)

[Out]

Timed out

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