Integrand size = 28, antiderivative size = 185 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (24+102 x^3-97 x^6\right )}{120 x^5 \left (-2+x^3\right )}+\frac {35 \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{12\ 2^{2/3} 3^{5/6}}-\frac {35 \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{36\ 2^{2/3} \sqrt [3]{3}}+\frac {35 \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{72\ 2^{2/3} \sqrt [3]{3}} \] Output:
1/120*(x^3+1)^(2/3)*(-97*x^6+102*x^3+24)/x^5/(x^3-2)+35/72*arctan(3^(5/6)* x/(3^(1/3)*x+2*2^(1/3)*(x^3+1)^(1/3)))*2^(1/3)*3^(1/6)-35/216*ln(-3*x+2^(1 /3)*3^(2/3)*(x^3+1)^(1/3))*2^(1/3)*3^(2/3)+35/432*ln(3*x^2+2^(1/3)*3^(2/3) *x*(x^3+1)^(1/3)+2^(2/3)*3^(1/3)*(x^3+1)^(2/3))*2^(1/3)*3^(2/3)
Time = 0.56 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {\frac {18 \left (1+x^3\right )^{2/3} \left (24+102 x^3-97 x^6\right )}{x^5 \left (-2+x^3\right )}+1050 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )-350 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )+175 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2160} \] Input:
Integrate[((1 + x^3)^(2/3)*(2 + x^3 + x^6))/(x^6*(-2 + x^3)^2),x]
Output:
((18*(1 + x^3)^(2/3)*(24 + 102*x^3 - 97*x^6))/(x^5*(-2 + x^3)) + 1050*2^(1 /3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 + x^3)^(1/3))] - 350*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3)] + 175*2^(1 /3)*3^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*3^(1/3 )*(1 + x^3)^(2/3)])/2160
Time = 0.65 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (x^6+x^3+2\right )}{x^6 \left (x^3-2\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 \left (x^3+1\right )^{2/3}}{4 \left (x^3-2\right )}+\frac {2 \left (x^3+1\right )^{2/3}}{\left (x^3-2\right )^2}+\frac {3 \left (x^3+1\right )^{2/3}}{4 x^3}+\frac {\left (x^3+1\right )^{2/3}}{2 x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\sqrt [3]{2} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3\ 3^{5/6}}+\frac {x \left (x^3+1\right )^{2/3}}{3 \left (2-x^3\right )}+\frac {\log \left (x^3-2\right )}{9\ 2^{2/3} \sqrt [3]{3}}+\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (x^3-2\right )-\frac {\log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3+1}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {3}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3+1}\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{8 x^2}\) |
Input:
Int[((1 + x^3)^(2/3)*(2 + x^3 + x^6))/(x^6*(-2 + x^3)^2),x]
Output:
(-3*(1 + x^3)^(2/3))/(8*x^2) + (x*(1 + x^3)^(2/3))/(3*(2 - x^3)) - (1 + x^ 3)^(5/3)/(10*x^5) + (2^(1/3)*ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(1 + x^3)^(1/ 3))/Sqrt[3]])/(3*3^(5/6)) + (3*3^(1/6)*ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(4*2^(2/3)) + ((3/2)^(2/3)*Log[-2 + x^3])/8 + Log[ -2 + x^3]/(9*2^(2/3)*3^(1/3)) - (3*(3/2)^(2/3)*Log[(3/2)^(1/3)*x - (1 + x^ 3)^(1/3)])/8 - Log[(3/2)^(1/3)*x - (1 + x^3)^(1/3)]/(3*2^(2/3)*3^(1/3))
Time = 13.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {\left (-1746 x^{6}+1836 x^{3}+432\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-350 \left (\left (\ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (x^{2}-x +1\right ) \left (1+x \right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (x^{2}-x +1\right ) \left (1+x \right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (x^{2}-x +1\right ) \left (1+x \right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-\frac {\ln \left (2\right )}{2}\right ) 3^{\frac {2}{3}}+3 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}\right ) 2^{\frac {1}{3}} x^{5} \left (x^{3}-2\right )}{2160 \left (x^{3}-2\right ) x^{5}}\) | \(177\) |
| trager | \(\text {Expression too large to display}\) | \(756\) |
| risch | \(\text {Expression too large to display}\) | \(920\) |
Input:
int((x^3+1)^(2/3)*(x^6+x^3+2)/x^6/(x^3-2)^2,x,method=_RETURNVERBOSE)
Output:
1/2160*((-1746*x^6+1836*x^3+432)*(x^3+1)^(2/3)-350*((ln((-2^(2/3)*3^(1/3)* x+2*((x^2-x+1)*(1+x))^(1/3))/x)-1/2*ln((2^(2/3)*3^(1/3)*((x^2-x+1)*(1+x))^ (1/3)*x+2^(1/3)*3^(2/3)*x^2+2*((x^2-x+1)*(1+x))^(2/3))/x^2)-1/2*ln(2))*3^( 2/3)+3*arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x^3+1)^(1/3)+3*x)/x)*3^(1/6) )*2^(1/3)*x^5*(x^3-2))/(x^3-2)/x^5
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (137) = 274\).
Time = 2.07 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.57 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=-\frac {350 \cdot 12^{\frac {2}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \log \left (\frac {18 \cdot 12^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 12^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 175 \cdot 12^{\frac {2}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \log \left (\frac {6 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 12^{\frac {1}{3}} {\left (55 \, x^{6} + 50 \, x^{3} + 4\right )} + 18 \, {\left (7 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 2100 \cdot 12^{\frac {1}{6}} {\left (x^{8} - 2 \, x^{5}\right )} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{7} - 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} + 600 \, x^{6} + 204 \, x^{3} + 8\right )} - 36 \, {\left (55 \, x^{8} + 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (487 \, x^{9} + 480 \, x^{6} + 12 \, x^{3} - 8\right )}}\right ) + 108 \, {\left (97 \, x^{6} - 102 \, x^{3} - 24\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{12960 \, {\left (x^{8} - 2 \, x^{5}\right )}} \] Input:
integrate((x^3+1)^(2/3)*(x^6+x^3+2)/x^6/(x^3-2)^2,x, algorithm="fricas")
Output:
-1/12960*(350*12^(2/3)*(x^8 - 2*x^5)*log((18*12^(1/3)*(x^3 + 1)^(1/3)*x^2 - 12^(2/3)*(x^3 - 2) - 36*(x^3 + 1)^(2/3)*x)/(x^3 - 2)) - 175*12^(2/3)*(x^ 8 - 2*x^5)*log((6*12^(2/3)*(4*x^4 + x)*(x^3 + 1)^(2/3) + 12^(1/3)*(55*x^6 + 50*x^3 + 4) + 18*(7*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 2 100*12^(1/6)*(x^8 - 2*x^5)*arctan(1/6*12^(1/6)*(12*12^(2/3)*(4*x^7 - 7*x^4 - 2*x)*(x^3 + 1)^(2/3) - 12^(1/3)*(377*x^9 + 600*x^6 + 204*x^3 + 8) - 36* (55*x^8 + 50*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(487*x^9 + 480*x^6 + 12*x^3 - 8 )) + 108*(97*x^6 - 102*x^3 - 24)*(x^3 + 1)^(2/3))/(x^8 - 2*x^5)
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((x**3+1)**(2/3)*(x**6+x**3+2)/x**6/(x**3-2)**2,x)
Output:
Timed out
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )}^{2} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^6+x^3+2)/x^6/(x^3-2)^2,x, algorithm="maxima")
Output:
integrate((x^6 + x^3 + 2)*(x^3 + 1)^(2/3)/((x^3 - 2)^2*x^6), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )}^{2} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^6+x^3+2)/x^6/(x^3-2)^2,x, algorithm="giac")
Output:
integrate((x^6 + x^3 + 2)*(x^3 + 1)^(2/3)/((x^3 - 2)^2*x^6), x)
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+x^3+2\right )}{x^6\,{\left (x^3-2\right )}^2} \,d x \] Input:
int(((x^3 + 1)^(2/3)*(x^3 + x^6 + 2))/(x^6*(x^3 - 2)^2),x)
Output:
int(((x^3 + 1)^(2/3)*(x^3 + x^6 + 2))/(x^6*(x^3 - 2)^2), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {-71 \left (x^{3}+1\right )^{\frac {2}{3}} x^{6}+136 \left (x^{3}+1\right )^{\frac {2}{3}} x^{3}+32 \left (x^{3}+1\right )^{\frac {2}{3}}+1050 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{9}-3 x^{6}+4}d x \right ) x^{8}-2100 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{9}-3 x^{6}+4}d x \right ) x^{5}}{160 x^{5} \left (x^{3}-2\right )} \] Input:
int((x^3+1)^(2/3)*(x^6+x^3+2)/x^6/(x^3-2)^2,x)
Output:
( - 71*(x**3 + 1)**(2/3)*x**6 + 136*(x**3 + 1)**(2/3)*x**3 + 32*(x**3 + 1) **(2/3) + 1050*int((x**3 + 1)**(2/3)/(x**9 - 3*x**6 + 4),x)*x**8 - 2100*in t((x**3 + 1)**(2/3)/(x**9 - 3*x**6 + 4),x)*x**5)/(160*x**5*(x**3 - 2))