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

Optimal. Leaf size=124 \[ -\log \left (\sqrt [3]{-x^7-x^4+2 x^3+x}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^7-x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{-x^7-x^4+2 x^3+x} x+\left (-x^7-x^4+2 x^3+x\right )^{2/3}\right ) \]

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Rubi [F]  time = 4.66, 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+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(((x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + x)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x
^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3))) - (12*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][
x^3/(1 + 2*x^6 - x^9 - x^18)^(2/3), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + ((1 - I*Sqrt[3]
)*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 + 2*x^6 - x^9 - x^18)^(2/3
)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + ((1 + I*Sqrt[3])*(x + 2*x^3 - x^4 - x^7)^(1/3)*
Defer[Subst][Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1
+ 2*x^2 - x^3 - x^6)^(1/3)) - (3*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((1 + x^3 + 2*x^6 +
x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) +
(12*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][x^3/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x
^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + (6*(x + 2*x^3 - x^4 - x^7)
^(1/3)*Defer[Subst][Defer[Int][x^6/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x],
x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + (15*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[I
nt][x^9/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 +
2*x^2 - x^3 - x^6)^(1/3)) + (3*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][x^12/((1 + x^3 + 2*x^6 +
x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3))

Rubi steps

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

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Mathematica [F]  time = 1.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.52, size = 124, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{-x^7-x^4+2 x^3+x}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^7-x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{-x^7-x^4+2 x^3+x} x+\left (-x^7-x^4+2 x^3+x\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.67, size = 174, normalized size = 1.40 \begin {gather*} \sqrt {3} \arctan \left (-\frac {70 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (32 \, x^{6} + 32 \, x^{3} - 39 \, x^{2} - 32\right )} - 56 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}}}{64 \, x^{6} + 64 \, x^{3} - 253 \, x^{2} - 64}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + x^{3} - x^{2} - 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{3} - x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*arctan(-(70*sqrt(3)*(-x^7 - x^4 + 2*x^3 + x)^(1/3)*x - sqrt(3)*(32*x^6 + 32*x^3 - 39*x^2 - 32) - 56*sq
rt(3)*(-x^7 - x^4 + 2*x^3 + x)^(2/3))/(64*x^6 + 64*x^3 - 253*x^2 - 64)) - 1/2*log((x^6 + x^3 - x^2 - 3*(-x^7 -
x^4 + 2*x^3 + x)^(1/3)*x + 3*(-x^7 - x^4 + 2*x^3 + x)^(2/3) - 1)/(x^6 + x^3 - 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^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + x^{3} + 2\right )}}{{\left (x^{6} + x^{3} - x^{2} - 1\right )} {\left (x^{6} + x^{3} - 2 \, x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 23.04, size = 536, normalized size = 4.32

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln((3161575138941971602132907799321044*RootOf(_Z^2-_Z+1)^2*x^6+43232446774215016867455325269137975*RootOf(_Z^
2-_Z+1)*x^6-50161601246391197894074705369999504*x^6+3161575138941971602132907799321044*RootOf(_Z^2-_Z+1)^2*x^3
-56117958716219995937859113437948531*RootOf(_Z^2-_Z+1)^2*x^2+43232446774215016867455325269137975*RootOf(_Z^2-_
Z+1)*x^3+99717198298490157965795846237779567*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(2/3)-6025233085750935052282
6993270182077*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(1/3)*x-29740930637917799974697972598107978*RootOf(_Z^2-_Z+
1)*x^2-50161601246391197894074705369999504*x^3-3161575138941971602132907799321044*RootOf(_Z^2-_Z+1)^2-60252330
857509350522826993270182077*(-x^7-x^4+2*x^3+x)^(2/3)-39464867440980807442968852967597490*x*(-x^7-x^4+2*x^3+x)^
(1/3)+96922415967603331524144345969151584*x^2-43232446774215016867455325269137975*RootOf(_Z^2-_Z+1)[phone]
6391197894074705369999504)/(-1+x)/(x^5+x^4+x^3+2*x^2+x+1))+RootOf(_Z^2-_Z+1)*ln((23921138623709266187215697152
6380*RootOf(_Z^2-_Z+1)^2*x^6-7168365858413273688491537072387909*RootOf(_Z^2-_Z+1)*x^6+535623877715702621580797
70140846928*x^6+239211386237092661872156971526380*RootOf(_Z^2-_Z+1)^2*x^3-4246002105708394748230786244593245*R
ootOf(_Z^2-_Z+1)^2*x^2-7168365858413273688491537072387909*RootOf(_Z^2-_Z+1)*x^3-997171982984901579657958462377
79567*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(2/3)+39464867440980807442968852967597490*RootOf(_Z^2-_Z+1)*(-x^7-x
^4+2*x^3+x)^(1/3)*x+71427487435393926297677159615636851*RootOf(_Z^2-_Z+1)*x^2+53562387771570262158079770140846
928*x^3-239211386237092661872156971526380*RootOf(_Z^2-_Z+1)^2+39464867440980807442968852967597490*(-x^7-x^4+2*
x^3+x)^(2/3)+60252330857509350522826993270182077*x*(-x^7-x^4+2*x^3+x)^(1/3)-1572863767895317222102342456516933
60*x^2+7168365858413273688491537072387909*RootOf(_Z^2-_Z+1)-53562387771570262158079770140846928)/(-1+x)/(x^5+x
^4+x^3+2*x^2+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^3 + 4*x^6 + 2)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((2*x^2 - x^3 - x^6 + 1)*(x^2 - x^3 - x^6 + 1)), 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^{6} + x^{3} - 2 x^{2} - 1\right )} \left (4 x^{6} + x^{3} + 2\right )}{\left (x - 1\right ) \left (x^{6} + x^{3} - 2 x^{2} - 1\right ) \left (x^{5} + x^{4} + x^{3} + 2 x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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