Integrand size = 47, antiderivative size = 238 \[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=-\frac {x \sqrt [3]{x-x^3+x^5+x^8}}{2 \left (2+x^2+2 x^4+2 x^7\right )}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{x-x^3+x^5+x^8}}{2^{2/3} \sqrt [3]{3} x-\sqrt [3]{x-x^3+x^5+x^8}}\right )}{6 \sqrt [3]{2} \sqrt [6]{3}}+\frac {\log \left (2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{x-x^3+x^5+x^8}\right )}{6 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (\sqrt [3]{2} 3^{2/3} x^2-2^{2/3} \sqrt [3]{3} x \sqrt [3]{x-x^3+x^5+x^8}+2 \left (x-x^3+x^5+x^8\right )^{2/3}\right )}{12 \sqrt [3]{2} 3^{2/3}} \]
-x*(x^8+x^5-x^3+x)^(1/3)/(4*x^7+4*x^4+2*x^2+4)+1/36*arctan(3^(1/2)*(x^8+x^ 5-x^3+x)^(1/3)/(2^(2/3)*3^(1/3)*x-(x^8+x^5-x^3+x)^(1/3)))*2^(2/3)*3^(5/6)+ 1/36*ln(2^(2/3)*3^(1/3)*x+2*(x^8+x^5-x^3+x)^(1/3))*2^(2/3)*3^(1/3)-1/72*ln (2^(1/3)*3^(2/3)*x^2-2^(2/3)*3^(1/3)*x*(x^8+x^5-x^3+x)^(1/3)+2*(x^8+x^5-x^ 3+x)^(2/3))*2^(2/3)*3^(1/3)
Time = 1.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=\frac {\sqrt [3]{x-x^3+x^5+x^8} \left (-\frac {36 x^{4/3}}{2+x^2+2 x^4+2 x^7}+\frac {2\ 2^{2/3} 3^{5/6} \arctan \left (\frac {3^{5/6} x^{2/3}}{\sqrt [3]{3} x^{2/3}-2 \sqrt [3]{2} \sqrt [3]{1-x^2+x^4+x^7}}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}+\frac {2\ 2^{2/3} \sqrt [3]{3} \log \left (3 x^{2/3}+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x^2+x^4+x^7}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}-\frac {2^{2/3} \sqrt [3]{3} \log \left (3 x^{4/3}-\sqrt [3]{2} 3^{2/3} x^{2/3} \sqrt [3]{1-x^2+x^4+x^7}+2^{2/3} \sqrt [3]{3} \left (1-x^2+x^4+x^7\right )^{2/3}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}\right )}{72 \sqrt [3]{x}} \]
((x - x^3 + x^5 + x^8)^(1/3)*((-36*x^(4/3))/(2 + x^2 + 2*x^4 + 2*x^7) + (2 *2^(2/3)*3^(5/6)*ArcTan[(3^(5/6)*x^(2/3))/(3^(1/3)*x^(2/3) - 2*2^(1/3)*(1 - x^2 + x^4 + x^7)^(1/3))])/(1 - x^2 + x^4 + x^7)^(1/3) + (2*2^(2/3)*3^(1/ 3)*Log[3*x^(2/3) + 2^(1/3)*3^(2/3)*(1 - x^2 + x^4 + x^7)^(1/3)])/(1 - x^2 + x^4 + x^7)^(1/3) - (2^(2/3)*3^(1/3)*Log[3*x^(4/3) - 2^(1/3)*3^(2/3)*x^(2 /3)*(1 - x^2 + x^4 + x^7)^(1/3) + 2^(2/3)*3^(1/3)*(1 - x^2 + x^4 + x^7)^(2 /3)])/(1 - x^2 + x^4 + x^7)^(1/3)))/(72*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.
| \(\displaystyle \int \frac {\left (5 x^7+2 x^4-2\right ) \sqrt [3]{x^8+x^5-x^3+x}}{\left (2 x^7+2 x^4+x^2+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^8+x^5-x^3+x} \int -\frac {\sqrt [3]{x} \left (-5 x^7-2 x^4+2\right ) \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}dx}{\sqrt [3]{x} \sqrt [3]{x^7+x^4-x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^8+x^5-x^3+x} \int \frac {\sqrt [3]{x} \left (-5 x^7-2 x^4+2\right ) \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}dx}{\sqrt [3]{x} \sqrt [3]{x^7+x^4-x^2+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^8+x^5-x^3+x} \int \frac {x \left (-5 x^7-2 x^4+2\right ) \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7+x^4-x^2+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^8+x^5-x^3+x} \int \left (\frac {x \left (6 x^4+5 x^2+14\right ) \sqrt [3]{x^7+x^4-x^2+1}}{2 \left (2 x^7+2 x^4+x^2+2\right )^2}-\frac {5 x \sqrt [3]{x^7+x^4-x^2+1}}{2 \left (2 x^7+2 x^4+x^2+2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7+x^4-x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^8+x^5-x^3+x} \left (7 \int \frac {x \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}d\sqrt [3]{x}-\frac {5}{2} \int \frac {x \sqrt [3]{x^7+x^4-x^2+1}}{2 x^7+2 x^4+x^2+2}d\sqrt [3]{x}+3 \int \frac {x^5 \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}d\sqrt [3]{x}+\frac {5}{2} \int \frac {x^3 \sqrt [3]{x^7+x^4-x^2+1}}{\left (2 x^7+2 x^4+x^2+2\right )^2}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{x^7+x^4-x^2+1}}\) |
3.27.66.3.1 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]
Timed out.
\[\int \frac {\left (5 x^{7}+2 x^{4}-2\right ) \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}}{\left (2 x^{7}+2 x^{4}+x^{2}+2\right )^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (188) = 376\).
Time = 34.00 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.67 \[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=-\frac {6 \cdot 18^{\frac {1}{6}} \sqrt {6} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \arctan \left (-\frac {18^{\frac {1}{6}} {\left (6 \cdot 18^{\frac {2}{3}} \sqrt {6} {\left (4 \, x^{15} + 8 \, x^{12} - 50 \, x^{10} + 4 \, x^{9} + 8 \, x^{8} - 50 \, x^{7} + 63 \, x^{5} - 50 \, x^{3} + 4 \, x\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} - 72 \, \sqrt {6} {\left (2 \, x^{14} + 4 \, x^{11} - 7 \, x^{9} + 2 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 7 \, x^{2} + 2\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} \sqrt {6} {\left (8 \, x^{21} + 24 \, x^{18} - 204 \, x^{16} + 24 \, x^{15} + 24 \, x^{14} - 408 \, x^{13} + 8 \, x^{12} + 648 \, x^{11} - 204 \, x^{10} - 408 \, x^{9} + 624 \, x^{8} + 24 \, x^{7} - 785 \, x^{6} + 624 \, x^{4} - 204 \, x^{2} + 8\right )}\right )}}{18 \, {\left (8 \, x^{21} + 24 \, x^{18} + 12 \, x^{16} + 24 \, x^{15} + 24 \, x^{14} + 24 \, x^{13} + 8 \, x^{12} - 432 \, x^{11} + 12 \, x^{10} + 24 \, x^{9} - 456 \, x^{8} + 24 \, x^{7} + 511 \, x^{6} - 456 \, x^{4} + 12 \, x^{2} + 8\right )}}\right ) + 18^{\frac {2}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \log \left (-\frac {36 \cdot 18^{\frac {1}{3}} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}} {\left (x^{7} + x^{4} - 4 \, x^{2} + 1\right )} - 18^{\frac {2}{3}} {\left (4 \, x^{14} + 8 \, x^{11} - 50 \, x^{9} + 4 \, x^{8} + 8 \, x^{7} - 50 \, x^{6} + 63 \, x^{4} - 50 \, x^{2} + 4\right )} - 54 \, {\left (4 \, x^{8} + 4 \, x^{5} - 7 \, x^{3} + 4 \, x\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}}}{4 \, x^{14} + 8 \, x^{11} + 4 \, x^{9} + 4 \, x^{8} + 8 \, x^{7} + 4 \, x^{6} + 9 \, x^{4} + 4 \, x^{2} + 4}\right ) - 2 \cdot 18^{\frac {2}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \log \left (\frac {3 \cdot 18^{\frac {2}{3}} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} x + 18^{\frac {1}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} + 18 \, {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}}}{2 \, x^{7} + 2 \, x^{4} + x^{2} + 2}\right ) + 324 \, {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} x}{648 \, {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )}} \]
-1/648*(6*18^(1/6)*sqrt(6)*(2*x^7 + 2*x^4 + x^2 + 2)*arctan(-1/18*18^(1/6) *(6*18^(2/3)*sqrt(6)*(4*x^15 + 8*x^12 - 50*x^10 + 4*x^9 + 8*x^8 - 50*x^7 + 63*x^5 - 50*x^3 + 4*x)*(x^8 + x^5 - x^3 + x)^(1/3) - 72*sqrt(6)*(2*x^14 + 4*x^11 - 7*x^9 + 2*x^8 + 4*x^7 - 7*x^6 - 7*x^2 + 2)*(x^8 + x^5 - x^3 + x) ^(2/3) + 18^(1/3)*sqrt(6)*(8*x^21 + 24*x^18 - 204*x^16 + 24*x^15 + 24*x^14 - 408*x^13 + 8*x^12 + 648*x^11 - 204*x^10 - 408*x^9 + 624*x^8 + 24*x^7 - 785*x^6 + 624*x^4 - 204*x^2 + 8))/(8*x^21 + 24*x^18 + 12*x^16 + 24*x^15 + 24*x^14 + 24*x^13 + 8*x^12 - 432*x^11 + 12*x^10 + 24*x^9 - 456*x^8 + 24*x^ 7 + 511*x^6 - 456*x^4 + 12*x^2 + 8)) + 18^(2/3)*(2*x^7 + 2*x^4 + x^2 + 2)* log(-(36*18^(1/3)*(x^8 + x^5 - x^3 + x)^(2/3)*(x^7 + x^4 - 4*x^2 + 1) - 18 ^(2/3)*(4*x^14 + 8*x^11 - 50*x^9 + 4*x^8 + 8*x^7 - 50*x^6 + 63*x^4 - 50*x^ 2 + 4) - 54*(4*x^8 + 4*x^5 - 7*x^3 + 4*x)*(x^8 + x^5 - x^3 + x)^(1/3))/(4* x^14 + 8*x^11 + 4*x^9 + 4*x^8 + 8*x^7 + 4*x^6 + 9*x^4 + 4*x^2 + 4)) - 2*18 ^(2/3)*(2*x^7 + 2*x^4 + x^2 + 2)*log((3*18^(2/3)*(x^8 + x^5 - x^3 + x)^(1/ 3)*x + 18^(1/3)*(2*x^7 + 2*x^4 + x^2 + 2) + 18*(x^8 + x^5 - x^3 + x)^(2/3) )/(2*x^7 + 2*x^4 + x^2 + 2)) + 324*(x^8 + x^5 - x^3 + x)^(1/3)*x)/(2*x^7 + 2*x^4 + x^2 + 2)
\[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=\int \frac {\sqrt [3]{x \left (x + 1\right ) \left (x^{6} - x^{5} + x^{4} - x + 1\right )} \left (5 x^{7} + 2 x^{4} - 2\right )}{\left (2 x^{7} + 2 x^{4} + x^{2} + 2\right )^{2}}\, dx \]
Integral((x*(x + 1)*(x**6 - x**5 + x**4 - x + 1))**(1/3)*(5*x**7 + 2*x**4 - 2)/(2*x**7 + 2*x**4 + x**2 + 2)**2, x)
\[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=\int { \frac {{\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2 \, x^{4} - 2\right )}}{{\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=\int { \frac {{\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2 \, x^{4} - 2\right )}}{{\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx=\int \frac {\left (5\,x^7+2\,x^4-2\right )\,{\left (x^8+x^5-x^3+x\right )}^{1/3}}{{\left (2\,x^7+2\,x^4+x^2+2\right )}^2} \,d x \]