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

Optimal. Leaf size=206 \[ \frac {1}{3} \log \left (\sqrt [3]{x^7+x^6-x^5+2 x^3-x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^7+x^6-x^5+2 x^3-x}}{\sqrt [3]{x^7+x^6-x^5+2 x^3-x}+2 x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^2+\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x+\left (x^7+x^6-x^5+2 x^3-x\right )^{2/3}\right )-\frac {\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x}{x^6+x^5-x^4+x^2-1} \]

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Rubi [F]  time = 4.53, 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 (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*Defer[Subst][Defer[Int][(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3)/(1 - x
^6 + x^12 - x^15 - x^18), x], x, x^(1/3)])/(x^(1/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) - (3*(-x + 2*x^3 - x
^5 + x^6 + x^7)^(1/3)*Defer[Subst][Defer[Int][(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3)/(-1 + x^6 - x^12 + x^15
+ x^18)^2, x], x, x^(1/3)])/(x^(1/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) + (18*(-x + 2*x^3 - x^5 + x^6 + x^7
)^(1/3)*Defer[Subst][Defer[Int][(x^3*(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18)^
2, x], x, x^(1/3)])/(x^(1/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) + (3*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*D
efer[Subst][Defer[Int][(x^6*(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18)^2, x], x,
x^(1/3)])/(x^(1/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) - (12*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*Defer[Sub
st][Defer[Int][(x^9*(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18)^2, x], x, x^(1/3)
])/(x^(1/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) - (3*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*Defer[Subst][Defer
[Int][(x^12*(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18)^2, x], x, x^(1/3)])/(x^(1
/3)*(-1 + 2*x^2 - x^4 + x^5 + x^6)^(1/3)) + (9*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x
^15*(-1 + 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18)^2, x], x, x^(1/3)])/(x^(1/3)*(-1
+ 2*x^2 - x^4 + x^5 + x^6)^(1/3)) + (12*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x^3*(-1
+ 2*x^6 - x^12 + x^15 + x^18)^(1/3))/(-1 + x^6 - x^12 + x^15 + x^18), x], x, x^(1/3)])/(x^(1/3)*(-1 + 2*x^2 -
x^4 + x^5 + x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx &=\frac {\sqrt [3]{-x+2 x^3-x^5+x^6+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6} \left (2-2 x^4+3 x^5+4 x^6\right )}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}} \left (2-2 x^{12}+3 x^{15}+4 x^{18}\right )}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (-1+6 x^3+x^6-4 x^9-x^{12}+3 x^{15}\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {\left (-1+4 x^3\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+6 x^3+x^6-4 x^9-x^{12}+3 x^{15}\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+4 x^3\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {6 x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {x^6 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}-\frac {4 x^9 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}-\frac {x^{12} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {3 x^{15} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{1-x^6+x^{12}-x^{15}-x^{18}}+\frac {4 x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{1-x^6+x^{12}-x^{15}-x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (9 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^{15} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (12 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^9 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (12 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (18 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ \end {align*}

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Mathematica [F]  time = 2.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.56, size = 206, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\sqrt [3]{x^7+x^6-x^5+2 x^3-x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^7+x^6-x^5+2 x^3-x}}{\sqrt [3]{x^7+x^6-x^5+2 x^3-x}+2 x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^2+\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x+\left (x^7+x^6-x^5+2 x^3-x\right )^{2/3}\right )-\frac {\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x}{x^6+x^5-x^4+x^2-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))/(-1 + x^2 - x^4 + x^5 + x^6)) - ArcTan[(Sqrt[3]*(-x + 2*x^3 - x^5 +
x^6 + x^7)^(1/3))/(2*x + (-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))]/Sqrt[3] + Log[-x + (-x + 2*x^3 - x^5 + x^6 +
x^7)^(1/3)]/3 - Log[x^2 + x*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3) + (-x + 2*x^3 - x^5 + x^6 + x^7)^(2/3)]/6

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fricas [A]  time = 11.41, size = 268, normalized size = 1.30 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}}}{3 \, {\left (x^{6} + x^{5} - x^{4} + 3 \, x^{2} - 1\right )}}\right ) + {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \log \left (\frac {x^{6} + x^{5} - x^{4} + x^{2} + 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{5} - x^{4} + x^{2} - 1}\right ) - 6 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x}{6 \, {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(3)*(x^6 + x^5 - x^4 + x^2 - 1)*arctan(-1/3*(2*sqrt(3)*(x^7 + x^6 - x^5 + 2*x^3 - x)^(1/3)*x + sqrt
(3)*(x^6 + x^5 - x^4 + x^2 - 1) - 2*sqrt(3)*(x^7 + x^6 - x^5 + 2*x^3 - x)^(2/3))/(x^6 + x^5 - x^4 + 3*x^2 - 1)
) + (x^6 + x^5 - x^4 + x^2 - 1)*log((x^6 + x^5 - x^4 + x^2 + 3*(x^7 + x^6 - x^5 + 2*x^3 - x)^(1/3)*x - 3*(x^7
+ x^6 - x^5 + 2*x^3 - x)^(2/3) - 1)/(x^6 + x^5 - x^4 + x^2 - 1)) - 6*(x^7 + x^6 - x^5 + 2*x^3 - x)^(1/3)*x)/(x
^6 + x^5 - 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^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + 3 \, x^{5} - 2 \, x^{4} + 2\right )}}{{\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 12.62, size = 3644, normalized size = 17.69 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^6+x^5-x^4+x^2-1)*(x*(x^6+x^5-x^4+2*x^2-1))^(1/3)+(1/6*RootOf(_Z^2+2*_Z+4)*ln(-(-852+372*(x^14+2*x^13-x^1
2-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^5-372*(x^14+2*x^13-x^12-2*x^11+
5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^4+744*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*
x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^2+372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8
-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^6-1704*x^11+46*RootOf(_Z^2+2*_Z+4)^2*x^4-46*RootOf(_Z^2+2*
_Z+4)^2*x^2-314*RootOf(_Z^2+2*_Z+4)*x^2-852*x^12-3692*x^7+852*x^10+1704*x^9+1704*x^5+5396*x^6+3692*x^2-4544*x^
8-5680*x^4-1044*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^6-1044*(x^14+2*x^13
-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^5+1044*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*
x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^4+372*RootOf(_Z^2+2*_Z+4)*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7
+6*x^6-4*x^4+x^2)^(2/3)-2088*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^2-372*
RootOf(_Z^2+2*_Z+4)*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)+23*RootOf(_Z^2+2*
_Z+4)^2*x^12+46*RootOf(_Z^2+2*_Z+4)^2*x^11-23*RootOf(_Z^2+2*_Z+4)^2*x^10-46*RootOf(_Z^2+2*_Z+4)^2*x^9+69*RootO
f(_Z^2+2*_Z+4)^2*x^8+46*RootOf(_Z^2+2*_Z+4)^2*x^7-92*RootOf(_Z^2+2*_Z+4)^2*x^6-46*RootOf(_Z^2+2*_Z+4)^2*x^5-4*
RootOf(_Z^2+2*_Z+4)-4*RootOf(_Z^2+2*_Z+4)*x^12-8*RootOf(_Z^2+2*_Z+4)*x^11+4*RootOf(_Z^2+2*_Z+4)*x^10+8*RootOf(
_Z^2+2*_Z+4)*x^9+314*RootOf(_Z^2+2*_Z+4)*x^7-306*RootOf(_Z^2+2*_Z+4)*x^6+8*RootOf(_Z^2+2*_Z+4)*x^5+23*RootOf(_
Z^2+2*_Z+4)^2+310*RootOf(_Z^2+2*_Z+4)*x^8+636*RootOf(_Z^2+2*_Z+4)*x^4+1044*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x
^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)-1044*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^
(2/3))/(1+x)/(x^5-x^3+x^2+x-1)/(x^6+x^5-x^4+x^2-1))-1/6*ln((752+372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^
8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^5-372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6
*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^4+744*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^
4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^2+372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1
/3)*RootOf(_Z^2+2*_Z+4)*x^6+1504*x^11-46*RootOf(_Z^2+2*_Z+4)^2*x^4+46*RootOf(_Z^2+2*_Z+4)^2*x^2-130*RootOf(_Z^
2+2*_Z+4)*x^2+752*x^12+4136*x^7-752*x^10-1504*x^9-1504*x^5-5640*x^6-4136*x^2+4888*x^8+6768*x^4+1788*(x^14+2*x^
13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^6+1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-
6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^5-1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)
^(1/3)*x^4+372*RootOf(_Z^2+2*_Z+4)*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(2/3)+35
76*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^2-372*RootOf(_Z^2+2*_Z+4)*(x^14+
2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)-23*RootOf(_Z^2+2*_Z+4)^2*x^12-46*RootOf(_Z^
2+2*_Z+4)^2*x^11+23*RootOf(_Z^2+2*_Z+4)^2*x^10+46*RootOf(_Z^2+2*_Z+4)^2*x^9-69*RootOf(_Z^2+2*_Z+4)^2*x^8-46*Ro
otOf(_Z^2+2*_Z+4)^2*x^7+92*RootOf(_Z^2+2*_Z+4)^2*x^6+46*RootOf(_Z^2+2*_Z+4)^2*x^5-96*RootOf(_Z^2+2*_Z+4)-96*Ro
otOf(_Z^2+2*_Z+4)*x^12-192*RootOf(_Z^2+2*_Z+4)*x^11+96*RootOf(_Z^2+2*_Z+4)*x^10+192*RootOf(_Z^2+2*_Z+4)*x^9+13
0*RootOf(_Z^2+2*_Z+4)*x^7+62*RootOf(_Z^2+2*_Z+4)*x^6+192*RootOf(_Z^2+2*_Z+4)*x^5-23*RootOf(_Z^2+2*_Z+4)^2+34*R
ootOf(_Z^2+2*_Z+4)*x^8+452*RootOf(_Z^2+2*_Z+4)*x^4-1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^
6-4*x^4+x^2)^(1/3)+1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(2/3))/(1+x)/(x^5-x
^3+x^2+x-1)/(x^6+x^5-x^4+x^2-1))*RootOf(_Z^2+2*_Z+4)-1/3*ln((752+372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x
^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^5-372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+
6*x^6-4*x^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^4+744*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x
^4+x^2)^(1/3)*RootOf(_Z^2+2*_Z+4)*x^2+372*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(
1/3)*RootOf(_Z^2+2*_Z+4)*x^6+1504*x^11-46*RootOf(_Z^2+2*_Z+4)^2*x^4+46*RootOf(_Z^2+2*_Z+4)^2*x^2-130*RootOf(_Z
^2+2*_Z+4)*x^2+752*x^12+4136*x^7-752*x^10-1504*x^9-1504*x^5-5640*x^6-4136*x^2+4888*x^8+6768*x^4+1788*(x^14+2*x
^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^6+1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9
-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^5-1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2
)^(1/3)*x^4+372*RootOf(_Z^2+2*_Z+4)*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(2/3)+3
576*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)*x^2-372*RootOf(_Z^2+2*_Z+4)*(x^14
+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(1/3)-23*RootOf(_Z^2+2*_Z+4)^2*x^12-46*RootOf(_Z
^2+2*_Z+4)^2*x^11+23*RootOf(_Z^2+2*_Z+4)^2*x^10+46*RootOf(_Z^2+2*_Z+4)^2*x^9-69*RootOf(_Z^2+2*_Z+4)^2*x^8-46*R
ootOf(_Z^2+2*_Z+4)^2*x^7+92*RootOf(_Z^2+2*_Z+4)^2*x^6+46*RootOf(_Z^2+2*_Z+4)^2*x^5-96*RootOf(_Z^2+2*_Z+4)-96*R
ootOf(_Z^2+2*_Z+4)*x^12-192*RootOf(_Z^2+2*_Z+4)*x^11+96*RootOf(_Z^2+2*_Z+4)*x^10+192*RootOf(_Z^2+2*_Z+4)*x^9+1
30*RootOf(_Z^2+2*_Z+4)*x^7+62*RootOf(_Z^2+2*_Z+4)*x^6+192*RootOf(_Z^2+2*_Z+4)*x^5-23*RootOf(_Z^2+2*_Z+4)^2+34*
RootOf(_Z^2+2*_Z+4)*x^8+452*RootOf(_Z^2+2*_Z+4)*x^4-1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x
^6-4*x^4+x^2)^(1/3)+1788*(x^14+2*x^13-x^12-2*x^11+5*x^10+4*x^9-6*x^8-2*x^7+6*x^6-4*x^4+x^2)^(2/3))/(1+x)/(x^5-
x^3+x^2+x-1)/(x^6+x^5-x^4+x^2-1)))*(x*(x^6+x^5-x^4+2*x^2-1))^(1/3)*(x^2*(x^6+x^5-x^4+2*x^2-1)^2)^(1/3)/x/(x^6+
x^5-x^4+2*x^2-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x*(x + 1)*(x**5 - x**3 + x**2 + x - 1))**(1/3)*(4*x**6 + 3*x**5 - 2*x**4 + 2)/(x**6 + x**5 - x**4 +
x**2 - 1)**2, x)

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