Integrand size = 30, antiderivative size = 214 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {2 \left (1+x^3\right )^{2/3} \left (-2+3 x^3\right )}{5 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^3}+2^{2/3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \] Output:
2/5*(x^3+1)^(2/3)*(3*x^3-2)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))* 3^(1/2)-5/6*arctan(3^(1/2)*x/(x+2*2^(1/3)*(x^3+1)^(1/3)))*2^(1/3)*3^(1/2)- 1/3*ln(-x+(x^3+1)^(1/3))+5/6*ln(-x+2^(1/3)*(x^3+1)^(1/3))*2^(1/3)+1/6*ln(x ^2+x*(x^3+1)^(1/3)+(x^3+1)^(2/3))-5/12*ln(x^2+2^(1/3)*x*(x^3+1)^(1/3)+2^(2 /3)*(x^3+1)^(2/3))*2^(1/3)
Time = 0.53 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {1}{60} \left (\frac {24 \left (1+x^3\right )^{2/3} \left (-2+3 x^3\right )}{x^5}+20 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )-20 \log \left (-x+\sqrt [3]{1+x^3}\right )+50 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^3}\right )+10 \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-25 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^3}+2^{2/3} \left (1+x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[((1 + x^3)^(2/3)*(8 - 4*x^3 + x^6))/(x^6*(2 + x^3)),x]
Output:
((24*(1 + x^3)^(2/3)*(-2 + 3*x^3))/x^5 + 20*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))] - 50*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/ 3)*(1 + x^3)^(1/3))] - 20*Log[-x + (1 + x^3)^(1/3)] + 50*2^(1/3)*Log[-x + 2^(1/3)*(1 + x^3)^(1/3)] + 10*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3 )] - 25*2^(1/3)*Log[x^2 + 2^(1/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*(1 + x^3)^(2 /3)])/60
Time = 0.76 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7276, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (x^6-4 x^3+8\right )}{x^6 \left (x^3+2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5 \left (x^3+1\right )^{2/3}}{x^3+2}-\frac {4 \left (x^3+1\right )^{2/3}}{x^3}+\frac {4 \left (x^3+1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5 \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (x^3+2\right )}{6\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3+1}\right )}{2\ 2^{2/3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {4 \left (x^3+1\right )^{5/3}}{5 x^5}+\frac {2 \left (x^3+1\right )^{2/3}}{x^2}\) |
Input:
Int[((1 + x^3)^(2/3)*(8 - 4*x^3 + x^6))/(x^6*(2 + x^3)),x]
Output:
(2*(1 + x^3)^(2/3))/x^2 - (4*(1 + x^3)^(5/3))/(5*x^5) + ArcTan[(1 + (2*x)/ (1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (5*ArcTan[(1 + (2^(2/3)*x)/(1 + x^3)^( 1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]) - (5*Log[2 + x^3])/(6*2^(2/3)) + (5*Log[ x/2^(1/3) - (1 + x^3)^(1/3)])/(2*2^(2/3)) - Log[-x + (1 + x^3)^(1/3)]/2
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]
Time = 3.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(\frac {50 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-25 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )+50 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-25 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-20 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+10 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+72 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-48 \left (x^{3}+1\right )^{\frac {2}{3}}}{60 x^{5}}\) | \(222\) |
Input:
int((x^3+1)^(2/3)*(x^6-4*x^3+8)/x^6/(x^3+2),x,method=_RETURNVERBOSE)
Output:
1/60*(50*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3+1)^(1/3))) *x^5-20*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3+1)^(1/3)))*x^5-25*2^(1/3)*x ^5*ln(2)+50*2^(1/3)*x^5*ln((-2^(2/3)*x+2*(x^3+1)^(1/3))/x)-25*2^(1/3)*x^5* ln((2^(2/3)*x*(x^3+1)^(1/3)+2^(1/3)*x^2+2*(x^3+1)^(2/3))/x^2)-20*ln((-x+(x ^3+1)^(1/3))/x)*x^5+10*ln((x^2+x*(x^3+1)^(1/3)+(x^3+1)^(2/3))/x^2)*x^5+72* x^3*(x^3+1)^(2/3)-48*(x^3+1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (165) = 330\).
Time = 14.76 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.66 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {300 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{7} + 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (91 \, x^{9} + 168 \, x^{6} + 84 \, x^{3} + 8\right )} + 12 \, {\left (19 \, x^{8} + 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{2 \, {\left (53 \, x^{9} + 48 \, x^{6} - 12 \, x^{3} - 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} + 2\right )} - 12 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} + 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) + 120 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 60 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 144 \, {\left (3 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, x^{5}} \] Input:
integrate((x^3+1)^(2/3)*(x^6-4*x^3+8)/x^6/(x^3+2),x, algorithm="fricas")
Output:
1/360*(300*4^(1/6)*sqrt(1/3)*x^5*arctan(1/2*4^(1/6)*sqrt(1/3)*(12*4^(2/3)* (2*x^7 + 5*x^4 + 2*x)*(x^3 + 1)^(2/3) + 4^(1/3)*(91*x^9 + 168*x^6 + 84*x^3 + 8) + 12*(19*x^8 + 22*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(53*x^9 + 48*x^6 - 1 2*x^3 - 8)) + 50*4^(2/3)*x^5*log(-(6*4^(1/3)*(x^3 + 1)^(1/3)*x^2 + 4^(2/3) *(x^3 + 2) - 12*(x^3 + 1)^(2/3)*x)/(x^3 + 2)) - 25*4^(2/3)*x^5*log((6*4^(2 /3)*(2*x^4 + x)*(x^3 + 1)^(2/3) + 4^(1/3)*(19*x^6 + 22*x^3 + 4) + 6*(5*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(x^6 + 4*x^3 + 4)) + 120*sqrt(3)*x^5*arctan(-(2 5382*sqrt(3)*(x^3 + 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt( 3)*(5831*x^3 + 7200))/(58653*x^3 + 8000)) - 60*x^5*log(3*(x^3 + 1)^(1/3)*x ^2 - 3*(x^3 + 1)^(2/3)*x + 1) + 144*(3*x^3 - 2)*(x^3 + 1)^(2/3))/x^5
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{2} + 2 x + 2\right ) \left (x^{4} - 2 x^{3} + 2 x^{2} - 4 x + 4\right )}{x^{6} \left (x^{3} + 2\right )}\, dx \] Input:
integrate((x**3+1)**(2/3)*(x**6-4*x**3+8)/x**6/(x**3+2),x)
Output:
Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(x**2 + 2*x + 2)*(x**4 - 2*x**3 + 2*x**2 - 4*x + 4)/(x**6*(x**3 + 2)), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - 4 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^6-4*x^3+8)/x^6/(x^3+2),x, algorithm="maxima")
Output:
integrate((x^6 - 4*x^3 + 8)*(x^3 + 1)^(2/3)/((x^3 + 2)*x^6), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - 4 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^6-4*x^3+8)/x^6/(x^3+2),x, algorithm="giac")
Output:
integrate((x^6 - 4*x^3 + 8)*(x^3 + 1)^(2/3)/((x^3 + 2)*x^6), x)
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-4\,x^3+8\right )}{x^6\,\left (x^3+2\right )} \,d x \] Input:
int(((x^3 + 1)^(2/3)*(x^6 - 4*x^3 + 8))/(x^6*(x^3 + 2)),x)
Output:
int(((x^3 + 1)^(2/3)*(x^6 - 4*x^3 + 8))/(x^6*(x^3 + 2)), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {16 \left (x^{3}+1\right )^{\frac {2}{3}} x^{3}-4 \left (x^{3}+1\right )^{\frac {2}{3}}+40 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{9}+3 x^{6}+2 x^{3}}d x \right ) x^{5}+5 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+2}d x \right ) x^{5}}{5 x^{5}} \] Input:
int((x^3+1)^(2/3)*(x^6-4*x^3+8)/x^6/(x^3+2),x)
Output:
(16*(x**3 + 1)**(2/3)*x**3 - 4*(x**3 + 1)**(2/3) + 40*int((x**3 + 1)**(2/3 )/(x**9 + 3*x**6 + 2*x**3),x)*x**5 + 5*int((x**3 + 1)**(2/3)/(x**3 + 2),x) *x**5)/(5*x**5)