\(\int \frac {(2-2 x^4+3 x^5+4 x^6) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{(-1+x^2-x^4+x^5+x^6)^2} \, dx\) [2494]

Optimal result

Integrand size = 60, antiderivative size = 206 \[ \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx=-\frac {x \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{-1+x^2-x^4+x^5+x^6}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{2 x+\sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}+\left (-x+2 x^3-x^5+x^6+x^7\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

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

Mathematica [A] (verified)

Time = 6.87 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.38 \[ \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx=\frac {\sqrt [3]{x \left (-1+2 x^2-x^4+x^5+x^6\right )} \left (-\frac {6 x^{4/3}}{-1+x^2-x^4+x^5+x^6}-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}{2 x^{2/3}+\sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\right )}{\sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {2 \log \left (-x^{2/3}+\sqrt [3]{-1+2 x^2-x^4+x^5+x^6}\right )}{\sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}+\left (-1+2 x^2-x^4+x^5+x^6\right )^{2/3}\right )}{\sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\right )}{6 \sqrt [3]{x}} \]

[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]

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

Maple [A] (verified)

Time = 17.94 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(\frac {\left (6 x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}+\left (x^{6}+x^{5}-x^{4}+x^{2}-1\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {x^{2}+x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}+\left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )\right ) x}{6 \left (x -\left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}\right ) \left (\left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}\right )\right )}\)	\(255\)
trager	\(-\frac {x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}}{x^{6}+x^{5}-x^{4}+x^{2}-1}+\frac {\ln \left (\frac {726439828539024 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-726439828539024 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}[phone] x^{6}-948727463035299 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}[phone] x^{5}-222287634496275 x^{4}-1131297147498108 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}} x +1695026266591056 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-726439828539024 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-47617244809781 \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}-329481804356255 x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}[phone] x^{2}-948727463035299 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-222287634496275}{x^{6}+x^{5}-x^{4}+x^{2}-1}\right )}{3}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {59576925050199 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-262005584529741 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-59576925050199 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-262005584529741 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}[phone] x^{6}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}[phone] x^{5}-148191756330850 x^{4}-1131297147498108 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}} x -139012825117131 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-59576925050199 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-329481804356255 \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}-47617244809781 x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}[phone] x^{2}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-148191756330850}{x^{6}+x^{5}-x^{4}+x^{2}-1}\right )\)	\(676\)
risch	\(\text {Expression too large to display}\)	\(3644\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 2.73 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.30 \[ \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}}}{3 \, {\left (x^{6} + x^{5} - x^{4} + 3 \, x^{2} - 1\right )}}\right ) + {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \log \left (\frac {x^{6} + x^{5} - x^{4} + x^{2} + 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{5} - x^{4} + x^{2} - 1}\right ) - 6 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x}{6 \, {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}} \]

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)