Integrand size = 21, antiderivative size = 168 \[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=-\frac {x^2 \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},x^3,-\frac {x^3}{2}\right )}{2 \sqrt [3]{2}}+\frac {2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\arctan \left (\frac {\sqrt [3]{3}+2 \sqrt [3]{2+x^3}}{3^{5/6}}\right )}{3^{5/6}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3}-\sqrt [3]{2+x^3}\right )}{2 \sqrt [3]{3}}-\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{2+x^3}\right )}{\sqrt [3]{3}} \]
-1/4*x^2*AppellF1(2/3,1,1/3,5/3,x^3,-1/2*x^3)*2^(2/3)+1/3*arctan(1/3*(3^(1 /3)+2*(x^3+2)^(1/3))*3^(1/6))*3^(1/6)+2/3*arctan(1/3*(1+2*3^(1/3)*x/(x^3+2 )^(1/3))*3^(1/2))*3^(1/6)+1/18*ln(-x^3+1)*3^(2/3)+1/6*ln(3^(1/3)-(x^3+2)^( 1/3))*3^(2/3)-1/3*ln(3^(1/3)*x-(x^3+2)^(1/3))*3^(2/3)
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]
Time = 0.34 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2583, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x+2}{\left (x^2+x+1\right ) \sqrt [3]{x^3+2}} \, dx\) |
\(\Big \downarrow \) 2583 |
\(\displaystyle \int \left (-\frac {x}{\left (1-x^3\right ) \sqrt [3]{x^3+2}}+\frac {2}{\left (1-x^3\right ) \sqrt [3]{x^3+2}}-\frac {x^2}{\left (1-x^3\right ) \sqrt [3]{x^3+2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x^2 \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},x^3,-\frac {x^3}{2}\right )}{2 \sqrt [3]{2}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\arctan \left (\frac {2 \sqrt [3]{x^3+2}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3}-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}-\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{\sqrt [3]{3}}\) |
-1/2*(x^2*AppellF1[2/3, 1, 1/3, 5/3, x^3, -1/2*x^3])/2^(1/3) + (2*ArcTan[( 1 + (2*3^(1/3)*x)/(2 + x^3)^(1/3))/Sqrt[3]])/3^(5/6) + ArcTan[(3^(1/3) + 2 *(2 + x^3)^(1/3))/3^(5/6)]/3^(5/6) + Log[1 - x^3]/(6*3^(1/3)) + Log[3^(1/3 ) - (2 + x^3)^(1/3)]/(2*3^(1/3)) - Log[3^(1/3)*x - (2 + x^3)^(1/3)]/3^(1/3 )
3.1.61.3.1 Defintions of rubi rules used
Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p _.), x_Symbol] :> Simp[1/c^q Int[ExpandIntegrand[(c^3 - d^3*x^3)^q*(a + b *x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Poly Q[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denomina tor[p], 3]
\[\int \frac {2+x}{\left (x^{2}+x +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x + 2}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {x + 2}{\sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x + 2}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x + 2}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {x+2}{{\left (x^3+2\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]
\[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {x}{\left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+\left (x^{3}+2\right )^{\frac {1}{3}} x +\left (x^{3}+2\right )^{\frac {1}{3}}}d x +2 \left (\int \frac {1}{\left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+\left (x^{3}+2\right )^{\frac {1}{3}} x +\left (x^{3}+2\right )^{\frac {1}{3}}}d x \right ) \]