Integrand size = 44, antiderivative size = 140 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{2\ 2^{2/3}} \]
3/10*(x^5+1)^(2/3)*(2*x^5+5*x^3+2)/x^5-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*2 ^(1/3)*(x^5+1)^(1/3)))*2^(1/3)+1/2*ln(-x+2^(1/3)*(x^5+1)^(1/3))*2^(1/3)-1/ 4*ln(x^2+2^(1/3)*x*(x^5+1)^(1/3)+2^(2/3)*(x^5+1)^(2/3))*2^(1/3)
Time = 2.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{2\ 2^{2/3}} \]
(3*(1 + x^5)^(2/3)*(2 + 5*x^3 + 2*x^5))/(10*x^5) - (Sqrt[3]*ArcTan[(Sqrt[3 ]*x)/(x + 2*2^(1/3)*(1 + x^5)^(1/3))])/2^(2/3) + Log[-x + 2^(1/3)*(1 + x^5 )^(1/3)]/2^(2/3) - Log[x^2 + 2^(1/3)*x*(1 + x^5)^(1/3) + 2^(2/3)*(1 + x^5) ^(2/3)]/(2*2^(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.
| \(\displaystyle \int \frac {\left (x^5+1\right )^{2/3} \left (2 x^5-3\right ) \left (2 x^5+x^3+2\right )}{x^6 \left (2 x^5-x^3+2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (x^5+1\right )^{2/3}}{x}-\frac {3 \left (x^5+1\right )^{2/3}}{x^6}-\frac {3 \left (x^5+1\right )^{2/3}}{x^3}+\frac {\left (x^5+1\right )^{2/3} \left (10 x^2-3\right )}{2 x^5-x^3+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {\left (x^5+1\right )^{2/3}}{2 x^5-x^3+2}dx+10 \int \frac {x^2 \left (x^5+1\right )^{2/3}}{2 x^5-x^3+2}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{5},\frac {3}{5},-x^5\right )}{2 x^2}+\frac {3 \left (x^5+1\right )^{2/3}}{5 x^5}+\frac {3}{5} \left (x^5+1\right )^{2/3}\) |
3.20.79.3.1 Defintions of rubi rules used
Time = 224.97 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.21
| method | result | size |
| pseudoelliptic | \(\frac {\left (12 x^{5}+30 x^{3}+12\right ) \left (x^{5}+1\right )^{\frac {2}{3}}+10 \,2^{\frac {1}{3}} x^{5} \left (\arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{5}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-\frac {\ln \left (\frac {2^{\frac {2}{3}} {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (2\right )}{2}\right )}{20 x^{5}}\) | \(169\) |
| risch | \(\text {Expression too large to display}\) | \(725\) |
| trager | \(\text {Expression too large to display}\) | \(995\) |
1/20*((12*x^5+30*x^3+12)*(x^5+1)^(2/3)+10*2^(1/3)*x^5*(arctan(1/3*3^(1/2)* (x+2*2^(1/3)*(x^5+1)^(1/3))/x)*3^(1/2)-1/2*ln((2^(2/3)*((1+x)*(x^4-x^3+x^2 -x+1))^(1/3)*x+2^(1/3)*x^2+2*((1+x)*(x^4-x^3+x^2-x+1))^(2/3))/x^2)+ln((-2^ (2/3)*x+2*((1+x)*(x^4-x^3+x^2-x+1))^(1/3))/x)-1/2*ln(2)))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (109) = 218\).
Time = 116.41 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.85 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=-\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \, {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )} - 12 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{5} - x^{3} + 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{5} + 5 \, x^{3} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{120 \, x^{5}} \]
-1/120*(20*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(2*x ^11 + x^9 - x^7 + 4*x^6 + x^4 + 2*x)*(x^5 + 1)^(2/3) + 4^(1/3)*(8*x^15 + 6 0*x^13 + 24*x^11 + 24*x^10 - x^9 + 120*x^8 + 24*x^6 + 24*x^5 + 60*x^3 + 8) + 12*(4*x^12 + 14*x^10 + x^8 + 8*x^7 + 14*x^5 + 4*x^2)*(x^5 + 1)^(1/3))/( 8*x^15 - 12*x^13 - 48*x^11 + 24*x^10 - x^9 - 24*x^8 - 48*x^6 + 24*x^5 - 12 *x^3 + 8)) - 10*4^(2/3)*x^5*log((6*4^(1/3)*(x^5 + 1)^(1/3)*x^2 + 4^(2/3)*( 2*x^5 - x^3 + 2) - 12*(x^5 + 1)^(2/3)*x)/(2*x^5 - x^3 + 2)) + 5*4^(2/3)*x^ 5*log((6*4^(2/3)*(x^6 + x^4 + x)*(x^5 + 1)^(2/3) + 4^(1/3)*(4*x^10 + 14*x^ 8 + x^6 + 8*x^5 + 14*x^3 + 4) + 6*(4*x^7 + x^5 + 4*x^2)*(x^5 + 1)^(1/3))/( 4*x^10 - 4*x^8 + x^6 + 8*x^5 - 4*x^3 + 4)) - 36*(2*x^5 + 5*x^3 + 2)*(x^5 + 1)^(2/3))/x^5
\[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{5} - 3\right ) \left (2 x^{5} + x^{3} + 2\right )}{x^{6} \cdot \left (2 x^{5} - x^{3} + 2\right )}\, dx \]
Integral(((x + 1)*(x**4 - x**3 + x**2 - x + 1))**(2/3)*(2*x**5 - 3)*(2*x** 5 + x**3 + 2)/(x**6*(2*x**5 - x**3 + 2)), x)
\[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} + 2\right )} {\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} + 2\right )} {\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx=\int \frac {{\left (x^5+1\right )}^{2/3}\,\left (2\,x^5-3\right )\,\left (2\,x^5+x^3+2\right )}{x^6\,\left (2\,x^5-x^3+2\right )} \,d x \]