Integrand size = 28, antiderivative size = 147 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^3}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.18 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-2 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+\log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )+\text {RootSum}\left [2-4 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{x^2 (1+x)}} \]
(x^(2/3)*(1 + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] - 2*Log[-x^(1/3) + (1 + x)^(1/3)] + Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)] + RootSum[2 - 4*#1^3 + 3*#1^6 & , (-(Log[x^(1 /3)]*#1^2) + Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^2)/(-2 + 3*#1^3) & ]))/(2* (x^2*(1 + x))^(1/3))
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2467, 2035, 7279, 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.
\(\displaystyle \int \frac {x^2+x+2}{\left (x^2+2 x+3\right ) \sqrt [3]{x^3+x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{x+1} \int \frac {x^2+x+2}{x^{2/3} \sqrt [3]{x+1} \left (x^2+2 x+3\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \int \frac {x^2+x+2}{\sqrt [3]{x+1} \left (x^2+2 x+3\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \int \left (\frac {1}{\sqrt [3]{x+1}}-\frac {(x+1)^{2/3}}{x^2+2 x+3}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {i \arctan \left (\frac {1+\frac {2 \sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}-i} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{2} \sqrt {3} \left (\sqrt {2}-i\right )^{2/3}}-\frac {i \arctan \left (\frac {1+\frac {2 \sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}+i} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{2} \sqrt {3} \left (\sqrt {2}+i\right )^{2/3}}-\frac {i \log \left (2 x+2 \left (1-i \sqrt {2}\right )\right )}{12 \sqrt [6]{2} \left (\sqrt {2}+i\right )^{2/3}}+\frac {i \log \left (2 x+2 \left (1+i \sqrt {2}\right )\right )}{12 \sqrt [6]{2} \left (\sqrt {2}-i\right )^{2/3}}-\frac {i \log \left (-\sqrt [3]{x+1}+\frac {\sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}-i}}\right )}{4 \sqrt [6]{2} \left (\sqrt {2}-i\right )^{2/3}}+\frac {i \log \left (-\sqrt [3]{x+1}+\frac {\sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}+i}}\right )}{4 \sqrt [6]{2} \left (\sqrt {2}+i\right )^{2/3}}-\frac {1}{2} \log \left (\sqrt [3]{x+1}-\sqrt [3]{x}\right )\right )}{\sqrt [3]{x^3+x^2}}\) |
(3*x^(2/3)*(1 + x)^(1/3)*(ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]]/ Sqrt[3] + ((I/2)*ArcTan[(1 + (2*2^(1/6)*x^(1/3))/((-I + Sqrt[2])^(1/3)*(1 + x)^(1/3)))/Sqrt[3]])/(2^(1/6)*Sqrt[3]*(-I + Sqrt[2])^(2/3)) - ((I/2)*Arc Tan[(1 + (2*2^(1/6)*x^(1/3))/((I + Sqrt[2])^(1/3)*(1 + x)^(1/3)))/Sqrt[3]] )/(2^(1/6)*Sqrt[3]*(I + Sqrt[2])^(2/3)) - ((I/12)*Log[2*(1 - I*Sqrt[2]) + 2*x])/(2^(1/6)*(I + Sqrt[2])^(2/3)) + ((I/12)*Log[2*(1 + I*Sqrt[2]) + 2*x] )/(2^(1/6)*(-I + Sqrt[2])^(2/3)) - ((I/4)*Log[(2^(1/6)*x^(1/3))/(-I + Sqrt [2])^(1/3) - (1 + x)^(1/3)])/(2^(1/6)*(-I + Sqrt[2])^(2/3)) + ((I/4)*Log[( 2^(1/6)*x^(1/3))/(I + Sqrt[2])^(1/3) - (1 + x)^(1/3)])/(2^(1/6)*(I + Sqrt[ 2])^(2/3)) - Log[-x^(1/3) + (1 + x)^(1/3)]/2))/(x^2 + x^3)^(1/3)
3.21.48.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 24.59 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{3 \textit {\_R}^{3}-2}\right )}{2}-\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )\) | \(130\) |
trager | \(\text {Expression too large to display}\) | \(14032\) |
1/2*ln(((x^2*(1+x))^(2/3)+(x^2*(1+x))^(1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3 *(2*(x^2*(1+x))^(1/3)+x)*3^(1/2)/x)+1/2*sum(_R^2*ln((-_R*x+(x^2*(1+x))^(1/ 3))/x)/(3*_R^3-2),_R=RootOf(3*_Z^6-4*_Z^3+2))-ln(((x^2*(1+x))^(1/3)-x)/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.29 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (i \, \sqrt {-3} x + i \, x\right )} - 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (-i \, \sqrt {-3} x + i \, x\right )} + 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (i \, \sqrt {-3} x - i \, x\right )} + 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (-i \, \sqrt {-3} x - i \, x\right )} - 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{36} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} \log \left (\frac {18^{\frac {1}{3}} {\left (-i \, \sqrt {2} x + 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 18 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{36} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} \log \left (\frac {18^{\frac {1}{3}} {\left (i \, \sqrt {2} x + 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 18 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
1/72*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*(sqrt(-3) - 1)*log((18^(1/3)*(sqrt(2)* (I*sqrt(-3)*x + I*x) - 2*sqrt(-3)*x - 2*x)*(I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3))/x) - 1/72*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*(sqrt(-3) + 1)*log ((18^(1/3)*(sqrt(2)*(-I*sqrt(-3)*x + I*x) + 2*sqrt(-3)*x - 2*x)*(I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3))/x) - 1/72*18^(2/3)*(-I*sqrt(2) - 4)^(1 /3)*(sqrt(-3) + 1)*log((18^(1/3)*(sqrt(2)*(I*sqrt(-3)*x - I*x) + 2*sqrt(-3 )*x - 2*x)*(-I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3))/x) + 1/72*18^(2/ 3)*(-I*sqrt(2) - 4)^(1/3)*(sqrt(-3) - 1)*log((18^(1/3)*(sqrt(2)*(-I*sqrt(- 3)*x - I*x) - 2*sqrt(-3)*x - 2*x)*(-I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^ (1/3))/x) + 1/36*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*log((18^(1/3)*(-I*sqrt(2)* x + 2*x)*(I*sqrt(2) - 4)^(2/3) + 18*(x^3 + x^2)^(1/3))/x) + 1/36*18^(2/3)* (-I*sqrt(2) - 4)^(1/3)*log((18^(1/3)*(I*sqrt(2)*x + 2*x)*(-I*sqrt(2) - 4)^ (2/3) + 18*(x^3 + x^2)^(1/3))/x) - sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt( 3)*(x^3 + x^2)^(1/3))/x) - log(-(x - (x^3 + x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)
Not integrable
Time = 2.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.18 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^{2} + x + 2}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{2} + 2 x + 3\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.19 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{2} + x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}} \,d x } \]
Exception generated. \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument Value-l n(abs((1/sageVARx+1)^(1/3)-1))+1/2*ln(((1/sageVARx+1)^(1/3))^2+(1/sageVARx +1)^(1/3)+1)
Not integrable
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.19 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^2+x+2}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^2+2\,x+3\right )} \,d x \]