Integrand size = 30, antiderivative size = 285 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]+2 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {2 \log (x) \text {$\#$1}-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ] \]
Time = 0.32 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-2 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+\log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )+4 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]-\frac {4}{3} \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-2 \log (x) \text {$\#$1}+6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}+\log (x) \text {$\#$1}^2-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2+2 \log (x) \text {$\#$1}^4-6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ]\right )}{2 \sqrt [3]{x^2 (1+x)}} \]
(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)] + 4*RootSum[-1 - #1 + #1^3 & , (-(Log[x^(1/3) ]*#1) + Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1)/(-1 + 3*#1^2) & ] - (4*RootSum [1 - #1 + #1^2 - 2*#1^3 + #1^4 + #1^6 & , (-2*Log[x]*#1 + 6*Log[(1 + x)^(1 /3) - x^(1/3)*#1]*#1 + Log[x]*#1^2 - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^ 2 + 2*Log[x]*#1^4 - 6*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^4)/(-1 + 2*#1 - 6 *#1^2 + 4*#1^3 + 6*#1^5) & ])/3))/(2*(x^2*(1 + x))^(1/3))
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.
\(\displaystyle \int \frac {x^3+x^2+1}{\left (x^3+x^2-1\right ) \sqrt [3]{x^3+x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{x+1} \int -\frac {x^3+x^2+1}{x^{2/3} \sqrt [3]{x+1} \left (-x^3-x^2+1\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {x^{2/3} \sqrt [3]{x+1} \int \frac {x^3+x^2+1}{x^{2/3} \sqrt [3]{x+1} \left (-x^3-x^2+1\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \frac {x^3+x^2+1}{\sqrt [3]{x+1} \left (-x^3-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \left (\frac {2}{\sqrt [3]{x+1} \left (-x^3-x^2+1\right )}-\frac {1}{\sqrt [3]{x+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \left (2 \int \frac {1}{\sqrt [3]{x+1} \left (-x^3-x^2+1\right )}d\sqrt [3]{x}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x+1}-\sqrt [3]{x}\right )\right )}{\sqrt [3]{x^3+x^2}}\) |
3.29.25.3.1 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]
Time = 110.89 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-\textit {\_Z} -1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{3 \textit {\_R}^{2}-1}\right )+\frac {\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R}^{3}+\textit {\_R} -2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{6 \textit {\_R}^{5}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )-\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )\) | \(200\) |
trager | \(\text {Expression too large to display}\) | \(486456\) |
2*sum(_R*ln((-_R*x+(x^2*(1+x))^(1/3))/x)/(3*_R^2-1),_R=RootOf(_Z^3-_Z-1))+ 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)+2*sum(_R*(2*_R^3+_R-2)*ln((-_R*x+(x^2* (1+x))^(1/3))/x)/(6*_R^5+4*_R^3-6*_R^2+2*_R-1),_R=RootOf(_Z^6+_Z^4-2*_Z^3+ _Z^2-_Z+1))-ln(((x^2*(1+x))^(1/3)-x)/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.57 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.69 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]
1/138*(3*sqrt(23)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100 /4761*sqrt(69) + 4/23)^(1/3))^2 - 16) - 69*(100/4761*sqrt(69) + 4/23)^(1/3 ) + 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log(1/69*(3*x*(69*(100/4761*sqrt(6 9) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 46*x*(69*(100/4 761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) + 3*(3*sq rt(23)*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/2 3)^(1/3)) - 46*sqrt(23)*x)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16) + 1656*x + 13800*(x^3 + x^2 )^(1/3))/x) - 1/138*(3*sqrt(23)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^ (1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16) + 69*(100/4761*sqrt(69 ) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log(1/69*(3*x*(69*(1 00/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 4 6*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1 /3)) - 3*(3*sqrt(23)*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761* sqrt(69) + 4/23)^(1/3)) - 46*sqrt(23)*x)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16) + 1656*x + 13 800*(x^3 + x^2)^(1/3))/x) + 1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/ (100/4761*sqrt(69) + 4/23)^(1/3))*log(-1/69*(3*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 46*x*(69*(100/4761*s qrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) + 1656*x - ...
Not integrable
Time = 13.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^{3} + x^{2} + 1}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{3} + x^{2} - 1\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}} \,d x } \]
Not integrable
Time = 1.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}} \,d x } \]
Not integrable
Time = 7.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^3+x^2+1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+x^2-1\right )} \,d x \]