Integrand size = 29, antiderivative size = 75 \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^6}}\right )-\log \left (-x+\sqrt [3]{x+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^6}+\left (x+x^6\right )^{2/3}\right ) \]
3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^6+x)^(1/3)))-ln(-x+(x^6+x)^(1/3))+1/2*ln( x^2+x*(x^6+x)^(1/3)+(x^6+x)^(2/3))
\[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx \]
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.
| \(\displaystyle \int \frac {2-3 x^5}{\left (x^5-x^2+1\right ) \sqrt [3]{x^6+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^5+1} \int \frac {2-3 x^5}{\sqrt [3]{x} \sqrt [3]{x^5+1} \left (x^5-x^2+1\right )}dx}{\sqrt [3]{x^6+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^5+1} \int \frac {\sqrt [3]{x} \left (2-3 x^5\right )}{\sqrt [3]{x^5+1} \left (x^5-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^6+x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^5+1} \int \left (\frac {\sqrt [3]{x} \left (5-3 x^2\right )}{\sqrt [3]{x^5+1} \left (x^5-x^2+1\right )}-\frac {3 \sqrt [3]{x}}{\sqrt [3]{x^5+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^6+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^5+1} \left (5 \int \frac {\sqrt [3]{x}}{\sqrt [3]{x^5+1} \left (x^5-x^2+1\right )}d\sqrt [3]{x}-3 \int \frac {x^{7/3}}{\sqrt [3]{x^5+1} \left (x^5-x^2+1\right )}d\sqrt [3]{x}-\frac {3}{2} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{15},\frac {1}{3},\frac {17}{15},-x^5\right )\right )}{\sqrt [3]{x^6+x}}\) |
3.10.93.3.1 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 = 6.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {-x +\left (x^{6}+x \right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}+x \left (x^{6}+x \right )^{\frac {1}{3}}+\left (x^{6}+x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}+x \right )^{\frac {1}{3}}\right )}{3 x}\right )\) | \(72\) |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {523628198354 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}+1588159912879 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+1081807032342 x^{5}-1047256396708 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+1553609277245 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}+1553609277245 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x +489077562720 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+523628198354 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+3228145779209 \left (x^{6}+x \right )^{\frac {2}{3}}+3228145779209 x \left (x^{6}+x \right )^{\frac {1}{3}}+1622710548513 x^{2}+1588159912879 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1081807032342}{x^{5}-x^{2}+1}\right )-\ln \left (\frac {8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-76335018 x^{5}-17923034 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -59851529 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92818507 \left (x^{6}+x \right )^{\frac {2}{3}}-92818507 x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 x^{2}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-76335018}{x^{5}-x^{2}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-76335018 x^{5}-17923034 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -59851529 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92818507 \left (x^{6}+x \right )^{\frac {2}{3}}-92818507 x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 x^{2}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-76335018}{x^{5}-x^{2}+1}\right )\) | \(522\) |
-ln((-x+(x^6+x)^(1/3))/x)+1/2*ln((x^2+x*(x^6+x)^(1/3)+(x^6+x)^(2/3))/x^2)- 3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^6+x)^(1/3)))
Time = 1.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {2}{3}}}{x^{5} + 8 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{6} + x\right )}^{\frac {2}{3}} + 1}{x^{5} - x^{2} + 1}\right ) \]
sqrt(3)*arctan(-(4*sqrt(3)*(x^6 + x)^(1/3)*x + sqrt(3)*(x^5 + 1) - 2*sqrt( 3)*(x^6 + x)^(2/3))/(x^5 + 8*x^2 + 1)) - 1/2*log((x^5 - x^2 + 3*(x^6 + x)^ (1/3)*x - 3*(x^6 + x)^(2/3) + 1)/(x^5 - x^2 + 1))
\[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=- \int \frac {3 x^{5}}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\, dx - \int \left (- \frac {2}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\right )\, dx \]
-Integral(3*x**5/(x**5*(x**6 + x)**(1/3) - x**2*(x**6 + x)**(1/3) + (x**6 + x)**(1/3)), x) - Integral(-2/(x**5*(x**6 + x)**(1/3) - x**2*(x**6 + x)** (1/3) + (x**6 + x)**(1/3)), x)
\[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\int { -\frac {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\int { -\frac {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=-\int \frac {3\,x^5-2}{{\left (x^6+x\right )}^{1/3}\,\left (x^5-x^2+1\right )} \,d x \]