3.13.77 \(\int \frac {(-1+x^4+2 x^6) \sqrt [3]{x+x^5+x^7}}{(1+x^4+x^6) (1-x^2+x^4+x^6)} \, dx\)

Optimal. Leaf size=92 \[ \frac {1}{2} \log \left (\sqrt [3]{x^7+x^5+x}-x\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^7+x^5+x}+x}\right )-\frac {1}{4} \log \left (x^2+\sqrt [3]{x^7+x^5+x} x+\left (x^7+x^5+x\right )^{2/3}\right ) \]

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Rubi [F]  time = 2.39, 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+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)),x]

[Out]

(3*(x + x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][x/(1 + x^6 + x^9)^(2/3), x], x, x^(2/3)])/(x^(1/3)*(1 + x^4 +
x^6)^(1/3)) - (9*(x + x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][x/((1 + x^6 + x^9)^(2/3)*(1 - x^3 + x^6 + x^9)
), x], x, x^(2/3)])/(2*x^(1/3)*(1 + x^4 + x^6)^(1/3)) + (3*(x + x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][x^4/(
(1 + x^6 + x^9)^(2/3)*(1 - x^3 + x^6 + x^9)), x], x, x^(2/3)])/(x^(1/3)*(1 + x^4 + x^6)^(1/3)) - (3*(x + x^5 +
x^7)^(1/3)*Defer[Subst][Defer[Int][x^7/((1 + x^6 + x^9)^(2/3)*(1 - x^3 + x^6 + x^9)), x], x, x^(2/3)])/(2*x^(
1/3)*(1 + x^4 + x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx &=\frac {\sqrt [3]{x+x^5+x^7} \int \frac {\sqrt [3]{x} \left (-1+x^4+2 x^6\right )}{\left (1+x^4+x^6\right )^{2/3} \left (1-x^2+x^4+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-1+x^{12}+2 x^{18}\right )}{\left (1+x^{12}+x^{18}\right )^{2/3} \left (1-x^6+x^{12}+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-1+x^6+2 x^9\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{\left (1+x^6+x^9\right )^{2/3}}+\frac {x \left (-3+2 x^3-x^6\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-3+2 x^3-x^6\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}+\frac {2 x^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}-\frac {x^7}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}-\frac {\left (9 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)),x]

[Out]

Integrate[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)), x]

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IntegrateAlgebraic [A]  time = 0.21, size = 92, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^5+x^7}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^5+x^7}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^5+x^7}+\left (x+x^5+x^7\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^5 + x^7)^(1/3))])/2 + Log[-x + (x + x^5 + x^7)^(1/3)]/2 - Log[x^2 +
x*(x + x^5 + x^7)^(1/3) + (x + x^5 + x^7)^(2/3)]/4

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fricas [A]  time = 3.21, size = 127, normalized size = 1.38 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{4} + x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{6} + x^{4} - x^{2} + 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} + x^{4} - x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+x^4-1)*(x^7+x^5+x)^(1/3)/(x^6+x^4+1)/(x^6+x^4-x^2+1),x, algorithm="fricas")

[Out]

1/2*sqrt(3)*arctan((2*sqrt(3)*(x^7 + x^5 + x)^(1/3)*x + sqrt(3)*(x^6 + x^4 + x^2 + 1) + 2*sqrt(3)*(x^7 + x^5 +
x)^(2/3))/(x^6 + x^4 - x^2 + 1)) + 1/4*log((x^6 + x^4 - x^2 + 3*(x^7 + x^5 + x)^(1/3)*x - 3*(x^7 + x^5 + x)^(
2/3) + 1)/(x^6 + x^4 - x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + x^{4} - 1\right )}}{{\left (x^{6} + x^{4} - x^{2} + 1\right )} {\left (x^{6} + x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+x^4-1)*(x^7+x^5+x)^(1/3)/(x^6+x^4+1)/(x^6+x^4-x^2+1),x, algorithm="giac")

[Out]

integrate((x^7 + x^5 + x)^(1/3)*(2*x^6 + x^4 - 1)/((x^6 + x^4 - x^2 + 1)*(x^6 + x^4 + 1)), x)

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maple [C]  time = 8.98, size = 729, normalized size = 7.92

method	result	size
trager	\(\frac {\ln \left (\frac {20906848965368 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}-23910827812306 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-11174168453619 x^{6}-23910827812306 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-62720546896104 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-11174168453619 x^{4}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}} x -64162034837974 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}[phone] \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-17963371600029 x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-14898891271492 x^{2}-23910827812306 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-11174168453619}{x^{6}+x^{4}-x^{2}+1}\right )}{2}-\frac {\ln \left (-\frac {14898891271492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}[phone] x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-44696673814476 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}[phone] x^{4}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-67166013684912 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}} x -23910827812306 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-15619635242427 \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-17963371600029 x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}[phone] x^{2}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )[phone]}{x^{6}+x^{4}-x^{2}+1}\right )}{2}-\ln \left (-\frac {14898891271492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}[phone] x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-44696673814476 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}[phone] x^{4}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-67166013684912 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}} x -23910827812306 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-15619635242427 \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-17963371600029 x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}[phone] x^{2}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )[phone]}{x^{6}+x^{4}-x^{2}+1}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\)	\(729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+x^4-1)*(x^7+x^5+x)^(1/3)/(x^6+x^4+1)/(x^6+x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*ln((20906848965368*RootOf(4*_Z^2+2*_Z+1)^2*x^6-23910827812306*RootOf(4*_Z^2+2*_Z+1)*x^6+20906848965368*Roo
tOf(4*_Z^2+2*_Z+1)^2*x^4-11174168453619*x^6-23910827812306*RootOf(4*_Z^2+2*_Z+1)*x^4-62720546896104*RootOf(4*_
Z^2+2*_Z+1)^2*x^2-11174168453619*x^4+35926743200058*RootOf(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(2/3)[phone]*Roo
tOf(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(1/3)*x-64162034837974*RootOf(4*_Z^2+2*_Z+1)*x^2+33583006842456*(x^7+x^5+x)^(2/
3)-17963371600029*x*(x^7+x^5+x)^(1/3)[phone]*RootOf(4*_Z^2+2*_Z+1)^2-14898891271492*x^2-23910827812306
*RootOf(4*_Z^2+2*_Z+1)-11174168453619)/(x^6+x^4-x^2+1))-1/2*ln(-(14898891271492*RootOf(4*_Z^2+2*_Z+1)^2*x^6+40
251207025668*RootOf(4*_Z^2+2*_Z+1)*x^6+14898891271492*RootOf(4*_Z^2+2*_Z+1)^2*x^4+20906848965368*x^6+402512070
25668*RootOf(4*_Z^2+2*_Z+1)*x^4-44696673814476*RootOf(4*_Z^2+2*_Z+1)^2*x^2+20906848965368*x^4+35926743200058*R
ootOf(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(2/3)-67166013684912*RootOf(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(1/3)*x-23910827812306
*RootOf(4*_Z^2+2*_Z+1)*x^2-15619635242427*(x^7+x^5+x)^(2/3)-17963371600029*x*(x^7+x^5+x)^(1/3)[phone]*
RootOf(4*_Z^2+2*_Z+1)^2+5226712241342*x^2+40251207025668*RootOf(4*_Z^2+2*_Z+1)[phone])/(x^6+x^4-x^2+1)
)-ln(-(14898891271492*RootOf(4*_Z^2+2*_Z+1)^2*x^6+40251207025668*RootOf(4*_Z^2+2*_Z+1)*x^6+14898891271492*Root
Of(4*_Z^2+2*_Z+1)^2*x^4+20906848965368*x^6+40251207025668*RootOf(4*_Z^2+2*_Z+1)*x^4-44696673814476*RootOf(4*_Z
^2+2*_Z+1)^2*x^2+20906848965368*x^4+35926743200058*RootOf(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(2/3)-67166013684912*Root
Of(4*_Z^2+2*_Z+1)*(x^7+x^5+x)^(1/3)*x-23910827812306*RootOf(4*_Z^2+2*_Z+1)*x^2-15619635242427*(x^7+x^5+x)^(2/3
)-17963371600029*x*(x^7+x^5+x)^(1/3)[phone]*RootOf(4*_Z^2+2*_Z+1)^2+5226712241342*x^2+40251207025668*R
ootOf(4*_Z^2+2*_Z+1)[phone])/(x^6+x^4-x^2+1))*RootOf(4*_Z^2+2*_Z+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + x^{4} - 1\right )}}{{\left (x^{6} + x^{4} - x^{2} + 1\right )} {\left (x^{6} + x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+x^4-1)*(x^7+x^5+x)^(1/3)/(x^6+x^4+1)/(x^6+x^4-x^2+1),x, algorithm="maxima")

[Out]

integrate((x^7 + x^5 + x)^(1/3)*(2*x^6 + x^4 - 1)/((x^6 + x^4 - x^2 + 1)*(x^6 + x^4 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6+x^4-1\right )\,{\left (x^7+x^5+x\right )}^{1/3}}{\left (x^6+x^4+1\right )\,\left (x^6+x^4-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 2*x^6 - 1)*(x + x^5 + x^7)^(1/3))/((x^4 + x^6 + 1)*(x^4 - x^2 + x^6 + 1)),x)

[Out]

int(((x^4 + 2*x^6 - 1)*(x + x^5 + x^7)^(1/3))/((x^4 + x^6 + 1)*(x^4 - x^2 + x^6 + 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**6+x**4-1)*(x**7+x**5+x)**(1/3)/(x**6+x**4+1)/(x**6+x**4-x**2+1),x)

[Out]

Timed out

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