\(\int \frac {(2+x)^2}{x (4-2 x+x^2) \sqrt [3]{1+x+x^2}} \, dx\) [1501]

Optimal result

Integrand size = 29, antiderivative size = 105 \[ \int \frac {(2+x)^2}{x \left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1+x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{1+x+x^2}+\left (1+x+x^2\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-3*((1 - I*Sqrt[3] + 2*x)/x)^(1/3)*((1 + I*Sqrt[3] + 2*x)/x)^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -1/2*(1 - I*S
qrt[3])/x, -1/2*(1 + I*Sqrt[3])/x])/(2*2^(2/3)*(1 + x + x^2)^(1/3)) + 6*Defer[Int][1/((4 - 2*x + x^2)*(1 + x +
x^2)^(1/3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt [3]{1+x+x^2}}+\frac {6}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}}\right ) \, dx \\ & = 6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx+\int \frac {1}{x \sqrt [3]{1+x+x^2}} \, dx \\ & = 6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {1+i \sqrt {3}+2 x}{x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+\frac {1}{2} \left (1-i \sqrt {3}\right ) x} \sqrt [3]{1+\frac {1}{2} \left (1+i \sqrt {3}\right ) x}} \, dx,x,\frac {1}{x}\right )}{2^{2/3} \left (\frac {1}{x}\right )^{2/3} \sqrt [3]{1+x+x^2}} \\ & = -\frac {3 \sqrt [3]{\frac {1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {1+i \sqrt {3}+2 x}{x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {1-i \sqrt {3}}{2 x},-\frac {1+i \sqrt {3}}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{1+x+x^2}}+6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {(2+x)^2}{x \left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {2-2 x+\sqrt [3]{1+x+x^2}}{\sqrt {3} \sqrt [3]{1+x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1+x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{1+x+x^2}+\left (1+x+x^2\right )^{2/3}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.21 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.55

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {2}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 x \left (x^{2}+x +1\right )^{\frac {2}{3}}-2 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}+x^{3}-\left (x^{2}+x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \left (x^{2}+x +1\right )^{\frac {2}{3}}+4 \left (x^{2}+x +1\right )^{\frac {1}{3}} x -4 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 \left (x^{2}+x +1\right )^{\frac {1}{3}}+2 x -2}{\left (x^{2}-2 x +4\right ) x}\right )-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}+x +1\right )^{\frac {2}{3}}+3 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{2}+x +1\right )^{\frac {2}{3}}-6 \left (x^{2}+x +1\right )^{\frac {1}{3}} x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{2}+x +1\right )^{\frac {1}{3}}+2 x +2}{\left (x^{2}-2 x +4\right ) x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}+x +1\right )^{\frac {2}{3}}+3 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{2}+x +1\right )^{\frac {2}{3}}-6 \left (x^{2}+x +1\right )^{\frac {1}{3}} x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{2}+x +1\right )^{\frac {1}{3}}+2 x +2}{\left (x^{2}-2 x +4\right ) x}\right )\)	\(583\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.32 \[ \int \frac {(2+x)^2}{x \left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{x^{3} - 11 \, x^{2} - 5 \, x - 9}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + 4 \, x}{x^{3} - 2 \, x^{2} + 4 \, x}\right ) \]

[In]

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

[Out]

-sqrt(3)*arctan((4*sqrt(3)*(x^2 + x + 1)^(2/3)*(x - 1) + 2*sqrt(3)*(x^2 + x + 1)^(1/3)*(x^2 - 2*x + 1) + sqrt(
3)*(x^3 - 3*x^2 + 3*x - 1))/(x^3 - 11*x^2 - 5*x - 9)) + 1/2*log((x^3 - 2*x^2 + 3*(x^2 + x + 1)^(2/3)*(x - 1) +
3*(x^2 + x + 1)^(1/3)*(x^2 - 2*x + 1) + 4*x)/(x^3 - 2*x^2 + 4*x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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