\(\int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} (1+3 x-2 x^2+x^3)} \, dx\) [1390]

Optimal result

Integrand size = 33, antiderivative size = 100 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{2+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2-2 x+\sqrt [3]{2+x^2}}{\sqrt {3} \sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 1.92 (sec) , antiderivative size = 511, normalized size of antiderivative = 5.11

method	result	size
trager	\(\ln \left (\frac {394 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-985 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x +318 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+797 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+1182 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-636 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -2682 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-891 \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+309 x^{3}+318 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+2391 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +891 \left (x^{2}+2\right )^{\frac {2}{3}}+1146 \left (x^{2}+2\right )^{\frac {1}{3}} x -1133 x^{2}-1379 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-573 \left (x^{2}+2\right )^{\frac {1}{3}}+927 x -721}{x^{3}-2 x^{2}+3 x +1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {206 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-515 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -891 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+291 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+618 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +573 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}+1782 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -367 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+318 \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-197 x^{3}-891 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+873 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -318 \left (x^{2}+2\right )^{\frac {2}{3}}+1146 \left (x^{2}+2\right )^{\frac {1}{3}} x +1182 x^{2}+721 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-573 \left (x^{2}+2\right )^{\frac {1}{3}}-591 x +1379}{x^{3}-2 x^{2}+3 x +1}\right )\)	\(511\)

[In]

int((x^2+2*x+6)/(x^2+2)^(1/3)/(x^3-2*x^2+3*x+1),x,method=_RETURNVERBOSE)

[Out]

ln((394*RootOf(_Z^2+_Z+1)^2*x^3-985*RootOf(_Z^2+_Z+1)^2*x^2-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)*x+318*RootOf(_
Z^2+_Z+1)*(x^2+2)^(1/3)*x^2+797*RootOf(_Z^2+_Z+1)*x^3+1182*RootOf(_Z^2+_Z+1)^2*x+573*RootOf(_Z^2+_Z+1)*(x^2+2)
^(2/3)-636*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x-2682*RootOf(_Z^2+_Z+1)*x^2-891*(x^2+2)^(2/3)*x-573*(x^2+2)^(1/3)*
x^2+309*x^3+318*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)+2391*RootOf(_Z^2+_Z+1)*x+891*(x^2+2)^(2/3)+1146*(x^2+2)^(1/3)*
x-1133*x^2-1379*RootOf(_Z^2+_Z+1)-573*(x^2+2)^(1/3)+927*x-721)/(x^3-2*x^2+3*x+1))+RootOf(_Z^2+_Z+1)*ln(-(206*R
ootOf(_Z^2+_Z+1)^2*x^3-515*RootOf(_Z^2+_Z+1)^2*x^2-573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)*x-891*RootOf(_Z^2+_Z+1)
*(x^2+2)^(1/3)*x^2+291*RootOf(_Z^2+_Z+1)*x^3+618*RootOf(_Z^2+_Z+1)^2*x+573*RootOf(_Z^2+_Z+1)*(x^2+2)^(2/3)+178
2*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)*x-367*RootOf(_Z^2+_Z+1)*x^2+318*(x^2+2)^(2/3)*x-573*(x^2+2)^(1/3)*x^2-197*x^
3-891*RootOf(_Z^2+_Z+1)*(x^2+2)^(1/3)+873*RootOf(_Z^2+_Z+1)*x-318*(x^2+2)^(2/3)+1146*(x^2+2)^(1/3)*x+1182*x^2+
721*RootOf(_Z^2+_Z+1)-573*(x^2+2)^(1/3)-591*x+1379)/(x^3-2*x^2+3*x+1))

Fricas [A] (verification not implemented)

none

Time = 0.71 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + 3 \, x + 1}{x^{3} - 2 \, x^{2} + 3 \, x + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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