Integrand size = 41, antiderivative size = 245 \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x-x^3+x^8}}\right )-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{-x-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{-x-x^3+x^8}\right )+\sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{-x-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3+x^8}+\left (-x-x^3+x^8\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{-x-x^3+x^8}-\sqrt [3]{2} \left (-x-x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^8-x^3-x)^(1/3)))-2^(1/3)*3^(1/2)*arctan( 3^(1/2)*x/(-x+2^(2/3)*(x^8-x^3-x)^(1/3)))-ln(x+(x^8-x^3-x)^(1/3))+2^(1/3)* ln(2*x+2^(2/3)*(x^8-x^3-x)^(1/3))+1/2*ln(x^2-x*(x^8-x^3-x)^(1/3)+(x^8-x^3- x)^(2/3))-1/2*ln(-2*x^2+2^(2/3)*x*(x^8-x^3-x)^(1/3)-2^(1/3)*(x^8-x^3-x)^(2 /3))*2^(1/3)
Time = 3.61 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.22 \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\frac {x^{2/3} \left (-1-x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2+x^7}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{-1-x^2+x^7}}\right )-2 \log \left (x^{2/3}+\sqrt [3]{-1-x^2+x^7}\right )+2 \sqrt [3]{2} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{-1-x^2+x^7}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^7}+\left (-1-x^2+x^7\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1-x^2+x^7}-\sqrt [3]{2} \left (-1-x^2+x^7\right )^{2/3}\right )\right )}{2 \left (x \left (-1-x^2+x^7\right )\right )^{2/3}} \]
(x^(2/3)*(-1 - x^2 + x^7)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2 /3) - 2*(-1 - x^2 + x^7)^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/ 3))/(x^(2/3) - 2^(2/3)*(-1 - x^2 + x^7)^(1/3))] - 2*Log[x^(2/3) + (-1 - x^ 2 + x^7)^(1/3)] + 2*2^(1/3)*Log[2*x^(2/3) + 2^(2/3)*(-1 - x^2 + x^7)^(1/3) ] + Log[x^(4/3) - x^(2/3)*(-1 - x^2 + x^7)^(1/3) + (-1 - x^2 + x^7)^(2/3)] - 2^(1/3)*Log[-2*x^(4/3) + 2^(2/3)*x^(2/3)*(-1 - x^2 + x^7)^(1/3) - 2^(1/ 3)*(-1 - x^2 + x^7)^(2/3)]))/(2*(x*(-1 - x^2 + x^7))^(2/3))
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.
| \(\displaystyle \int \frac {\left (5 x^7+2\right ) \sqrt [3]{x^8-x^3-x}}{\left (x^7-1\right ) \left (x^7+x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^8-x^3-x} \int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1} \left (5 x^7+2\right )}{\left (1-x^7\right ) \left (-x^7-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \int \frac {x \sqrt [3]{x^7-x^2-1} \left (5 x^7+2\right )}{\left (1-x^7\right ) \left (-x^7-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \int \left (\frac {\sqrt [3]{x^7-x^2-1} \left (-\sqrt [3]{x}-2\right )}{3 \left (x^{2/3}+\sqrt [3]{x}+1\right )}+\frac {\sqrt [3]{x^7-x^2-1}}{3 \left (\sqrt [3]{x}-1\right )}+\frac {\left (-x^{5/3}-2 x^{4/3}+4 x+3 x^{2/3}+2 \sqrt [3]{x}+1\right ) \sqrt [3]{x^7-x^2-1}}{3 \left (x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1\right )}+\frac {\left (x^{11/3}+11 x^3-10 x^{8/3}+8 x^2-7 x^{5/3}+x^{4/3}+5 x-3 \sqrt [3]{x}+2\right ) \sqrt [3]{x^7-x^2-1}}{3 \left (x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1\right )}-\frac {x \left (7 x^5+2\right ) \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \left (\frac {1}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{\sqrt [3]{x}-1}d\sqrt [3]{x}-\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^7-x^2-1}}{2 \sqrt [3]{x}-i \sqrt {3}+1}d\sqrt [3]{x}-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^7-x^2-1}}{2 \sqrt [3]{x}+i \sqrt {3}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {2}{3} \int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\int \frac {x^{2/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {4}{3} \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {2}{3} \int \frac {x^{4/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{3} \int \frac {x^{5/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {2}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {5}{3} \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {x^{4/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {7}{3} \int \frac {x^{5/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {8}{3} \int \frac {x^2 \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {10}{3} \int \frac {x^{8/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {11}{3} \int \frac {x^3 \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {x^{11/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-2 \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}d\sqrt [3]{x}-7 \int \frac {x^6 \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\) |
3.27.97.3.1 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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {\left (5 x^{7}+2\right ) \left (x^{8}-x^{3}-x \right )^{\frac {1}{3}}}{\left (x^{7}-1\right ) \left (x^{7}+x^{2}-1\right )}d x\]
Exception generated. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Timed out. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int { \frac {{\left (x^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}} \,d x } \]
\[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int { \frac {{\left (x^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int \frac {\left (5\,x^7+2\right )\,{\left (x^8-x^3-x\right )}^{1/3}}{\left (x^7-1\right )\,\left (x^7+x^2-1\right )} \,d x \]