Integrand size = 39, antiderivative size = 161 \[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{2 x^2}-3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^2+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^2+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^2+2 x^3}+\sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \]
3/2*(2*x^3+2*x^2+1)^(2/3)/x^2-3*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*(2*x ^3+2*x^2+1)^(1/3)))+3^(2/3)*ln(-3*x+3^(2/3)*(2*x^3+2*x^2+1)^(1/3))-1/2*3^( 2/3)*ln(3*x^2+3^(2/3)*x*(2*x^3+2*x^2+1)^(1/3)+3^(1/3)*(2*x^3+2*x^2+1)^(2/3 ))
Time = 0.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{2 x^2}-3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^2+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^2+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^2+2 x^3}+\sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \]
(3*(1 + 2*x^2 + 2*x^3)^(2/3))/(2*x^2) - 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1 /3)*x + 2*(1 + 2*x^2 + 2*x^3)^(1/3))] + 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + 2* x^2 + 2*x^3)^(1/3)] - (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + 2*x^2 + 2*x^3)^( 1/3) + 3^(1/3)*(1 + 2*x^2 + 2*x^3)^(2/3)])/2
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.
| \(\displaystyle \int \frac {\left (2 x^2+3\right ) \left (2 x^3+2 x^2+1\right )^{2/3}}{x^3 \left (x^3-2 x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (2 x^3+2 x^2+1\right )^{2/3} \left (-4 x^2+8 x+3\right )}{x^3-2 x^2-1}+\frac {4 \left (2 x^3+2 x^2+1\right )^{2/3}}{x}-\frac {3 \left (2 x^3+2 x^2+1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {27\ 2^{2/3} \left (2 x^3+2 x^2+1\right )^{2/3} \text {Subst}\left (\int \frac {\left (2 x+\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )\right )^{2/3} \left (4 x^2-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )\right )^{2/3}}{\left (x-\frac {1}{3}\right )^3}dx,x,x+\frac {1}{3}\right )}{\left (12 x+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+4\right )^{2/3} \left (4 (3 x+1)^2-2 \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) (3 x+1)+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}+\left (62-6 \sqrt {105}\right )^{2/3}-4\right )^{2/3}}+\frac {36\ 2^{2/3} \left (2 x^3+2 x^2+1\right )^{2/3} \text {Subst}\left (\int \frac {\left (2 x+\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )\right )^{2/3} \left (4 x^2-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )\right )^{2/3}}{x-\frac {1}{3}}dx,x,x+\frac {1}{3}\right )}{\left (12 x+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+4\right )^{2/3} \left (4 (3 x+1)^2-2 \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) (3 x+1)+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}+\left (62-6 \sqrt {105}\right )^{2/3}-4\right )^{2/3}}+3 \int \frac {\left (2 x^3+2 x^2+1\right )^{2/3}}{x^3-2 x^2-1}dx+8 \int \frac {x \left (2 x^3+2 x^2+1\right )^{2/3}}{x^3-2 x^2-1}dx-4 \int \frac {x^2 \left (2 x^3+2 x^2+1\right )^{2/3}}{x^3-2 x^2-1}dx\) |
3.22.75.3.1 Defintions of rubi rules used
Time = 16.67 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {2 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}} x +\left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+6 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{2}+3 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(151\) |
| risch | \(\text {Expression too large to display}\) | \(749\) |
| trager | \(\text {Expression too large to display}\) | \(1041\) |
1/2*(2*3^(2/3)*ln((-3^(1/3)*x+(2*x^3+2*x^2+1)^(1/3))/x)*x^2-3^(2/3)*ln((3^ (2/3)*x^2+3^(1/3)*(2*x^3+2*x^2+1)^(1/3)*x+(2*x^3+2*x^2+1)^(2/3))/x^2)*x^2+ 6*3^(1/6)*arctan(1/9*3^(1/2)*(2*3^(2/3)*(2*x^3+2*x^2+1)^(1/3)+3*x)/x)*x^2+ 3*(2*x^3+2*x^2+1)^(2/3))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (131) = 262\).
Time = 10.98 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.72 \[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=-\frac {2 \cdot 9^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {2 \cdot 9^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{7} - 14 \, x^{6} - 4 \, x^{5} - 7 \, x^{4} - 4 \, x^{3} - x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} - 6 \cdot 9^{\frac {1}{3}} \sqrt {3} {\left (55 \, x^{8} + 50 \, x^{7} + 4 \, x^{6} + 25 \, x^{5} + 4 \, x^{4} + x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (377 \, x^{9} + 600 \, x^{8} + 204 \, x^{7} + 308 \, x^{6} + 204 \, x^{5} + 12 \, x^{4} + 51 \, x^{3} + 6 \, x^{2} + 1\right )}}{3 \, {\left (487 \, x^{9} + 480 \, x^{8} + 12 \, x^{7} + 232 \, x^{6} + 12 \, x^{5} - 12 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} - 1\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} x^{2} \log \left (\frac {3 \cdot 9^{\frac {2}{3}} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} x - 9^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} - 1\right )}}{x^{3} - 2 \, x^{2} - 1}\right ) + 9^{\frac {1}{3}} x^{2} \log \left (\frac {9 \cdot 9^{\frac {1}{3}} {\left (8 \, x^{4} + 2 \, x^{3} + x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} {\left (55 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} + 25 \, x^{3} + 4 \, x^{2} + 1\right )} + 27 \, {\left (7 \, x^{5} + 4 \, x^{4} + 2 \, x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
-1/6*(2*9^(1/3)*sqrt(3)*x^2*arctan(1/3*(2*9^(2/3)*sqrt(3)*(8*x^7 - 14*x^6 - 4*x^5 - 7*x^4 - 4*x^3 - x)*(2*x^3 + 2*x^2 + 1)^(2/3) - 6*9^(1/3)*sqrt(3) *(55*x^8 + 50*x^7 + 4*x^6 + 25*x^5 + 4*x^4 + x^2)*(2*x^3 + 2*x^2 + 1)^(1/3 ) - sqrt(3)*(377*x^9 + 600*x^8 + 204*x^7 + 308*x^6 + 204*x^5 + 12*x^4 + 51 *x^3 + 6*x^2 + 1))/(487*x^9 + 480*x^8 + 12*x^7 + 232*x^6 + 12*x^5 - 12*x^4 + 3*x^3 - 6*x^2 - 1)) - 2*9^(1/3)*x^2*log((3*9^(2/3)*(2*x^3 + 2*x^2 + 1)^ (1/3)*x^2 - 9*(2*x^3 + 2*x^2 + 1)^(2/3)*x - 9^(1/3)*(x^3 - 2*x^2 - 1))/(x^ 3 - 2*x^2 - 1)) + 9^(1/3)*x^2*log((9*9^(1/3)*(8*x^4 + 2*x^3 + x)*(2*x^3 + 2*x^2 + 1)^(2/3) + 9^(2/3)*(55*x^6 + 50*x^5 + 4*x^4 + 25*x^3 + 4*x^2 + 1) + 27*(7*x^5 + 4*x^4 + 2*x^2)*(2*x^3 + 2*x^2 + 1)^(1/3))/(x^6 - 4*x^5 + 4*x ^4 - 2*x^3 + 4*x^2 + 1)) - 9*(2*x^3 + 2*x^2 + 1)^(2/3))/x^2
\[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\int \frac {\left (2 x^{2} + 3\right ) \left (2 x^{3} + 2 x^{2} + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} - 2 x^{2} - 1\right )}\, dx \]
\[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{2} + 3\right )}}{{\left (x^{3} - 2 \, x^{2} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{2} + 3\right )}}{{\left (x^{3} - 2 \, x^{2} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx=\int -\frac {\left (2\,x^2+3\right )\,{\left (2\,x^3+2\,x^2+1\right )}^{2/3}}{x^3\,\left (-x^3+2\,x^2+1\right )} \,d x \]