\(\int \frac {(2+x^3+4 x^6) \sqrt [3]{x+2 x^3-x^4-x^7}}{(-1-2 x^2+x^3+x^6) (-1-x^2+x^3+x^6)} \, dx\) [1827]

Optimal result

Integrand size = 62, antiderivative size = 124 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+2 x^3-x^4-x^7}}\right )-\log \left (-x+\sqrt [3]{x+2 x^3-x^4-x^7}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+2 x^3-x^4-x^7}+\left (x+2 x^3-x^4-x^7\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(x+2*(-x^7-x^4+2*x^3+x)^(1/3)))-ln(-x+(-x^7-x^4+2*x^3+x)^(1/3))+1/2*ln(x^2+x*(-x^7-x
^4+2*x^3+x)^(1/3)+(-x^7-x^4+2*x^3+x)^(2/3))

Rubi [F]

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

[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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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} \text {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 ) \text {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 ) \text {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} \text {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} \text {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 ) \text {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 ) \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\frac {x^{2/3} \left (-1-2 x^2+x^3+x^6\right )^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-2 x^2+x^3+x^6}}\right )+2 \log \left (x^{2/3}+\sqrt [3]{-1-2 x^2+x^3+x^6}\right )-\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-2 x^2+x^3+x^6}+\left (-1-2 x^2+x^3+x^6\right )^{2/3}\right )\right )}{2 \left (-x \left (-1-2 x^2+x^3+x^6\right )\right )^{2/3}} \]

[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]

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

Maple [A] (verified)

Time = 15.78 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {2}{3}}+{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\)	\(115\)
trager	\(\text {Expression too large to display}\)	\(536\)

[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,method=_RETURNVERBOSE)

[Out]

1/2*ln(((-x*(x^6+x^3-2*x^2-1))^(2/3)+(-x*(x^6+x^3-2*x^2-1))^(1/3)*x+x^2)/x^2)+3^(1/2)*arctan(1/3*(2*(-x*(x^6+x
^3-2*x^2-1))^(1/3)+x)*3^(1/2)/x)-ln(((-x*(x^6+x^3-2*x^2-1))^(1/3)-x)/x)

Fricas [A] (verification not implemented)

none

Time = 1.65 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\sqrt {3} \arctan \left (-\frac {70 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (32 \, x^{6} + 32 \, x^{3} - 39 \, x^{2} - 32\right )} - 56 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}}}{64 \, x^{6} + 64 \, x^{3} - 253 \, x^{2} - 64}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + x^{3} - x^{2} - 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{3} - x^{2} - 1}\right ) \]

[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))

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\left (4\,x^6+x^3+2\right )\,{\left (-x^7-x^4+2\,x^3+x\right )}^{1/3}}{\left (-x^6-x^3+2\,x^2+1\right )\,\left (-x^6-x^3+x^2+1\right )} \,d x \]

[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)