Integrand size = 29, antiderivative size = 141 \[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=-\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
-3/2*(3*x^3+x+1)^(2/3)/x^2+2^(2/3)*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(3* x^3+x+1)^(1/3)))-2^(2/3)*ln(-2*x+2^(2/3)*(3*x^3+x+1)^(1/3))+1/2*ln(2*x^2+2 ^(2/3)*x*(3*x^3+x+1)^(1/3)+2^(1/3)*(3*x^3+x+1)^(2/3))*2^(2/3)
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=-\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
(-3*(1 + x + 3*x^3)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x + 3*x^3)^(1/3))] - 2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x + 3 *x^3)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(1 + x + 3*x^3)^(1/3) + 2^(1/3)*(1 + x + 3*x^3)^(2/3)]/2^(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.
| \(\displaystyle \int \frac {(2 x+3) \left (3 x^3+x+1\right )^{2/3}}{x^3 \left (x^3+x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (3 x^3+x+1\right )^{2/3}}{x}+\frac {3 \left (3 x^3+x+1\right )^{2/3}}{x^3}+\frac {\left (3 x^3+x+1\right )^{2/3} \left (-x^2+x-4\right )}{x^3+x+1}-\frac {\left (3 x^3+x+1\right )^{2/3}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (3 x^3+x+1\right )^{2/3} \int \frac {\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}{x^3}dx}{\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}-\frac {\left (3 x^3+x+1\right )^{2/3} \int \frac {\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}{x^2}dx}{\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}+\frac {\left (3 x^3+x+1\right )^{2/3} \int \frac {\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}{x}dx}{\left (3 x-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+\sqrt [3]{\frac {2}{-9+\sqrt {85}}}\right )^{2/3} \left (9 x^2-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+1\right )^{2/3}}-4 \int \frac {\left (3 x^3+x+1\right )^{2/3}}{x^3+x+1}dx+\int \frac {x \left (3 x^3+x+1\right )^{2/3}}{x^3+x+1}dx-\int \frac {x^2 \left (3 x^3+x+1\right )^{2/3}}{x^3+x+1}dx\) |
3.20.96.3.1 Defintions of rubi rules used
Time = 15.46 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (3 x^{3}+x +1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (3 x^{3}+x +1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (3 x^{3}+x +1\right )^{\frac {1}{3}} x +\left (3 x^{3}+x +1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-3 \left (3 x^{3}+x +1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(130\) |
| risch | \(\text {Expression too large to display}\) | \(935\) |
| trager | \(\text {Expression too large to display}\) | \(1473\) |
1/2*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(3*x^3+x+1)^(1/3)) )*x^2-2*2^(2/3)*ln((-2^(1/3)*x+(3*x^3+x+1)^(1/3))/x)*x^2+2^(2/3)*ln((2^(2/ 3)*x^2+2^(1/3)*(3*x^3+x+1)^(1/3)*x+(3*x^3+x+1)^(2/3))/x^2)*x^2-3*(3*x^3+x+ 1)^(2/3))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (112) = 224\).
Time = 8.10 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.70 \[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=\frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{5} + 8 \, x^{4} + x^{3} + 2 \, x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{6} + 20 \, x^{5} + x^{4} + 2 \, x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{7} + 255 \, x^{6} + 39 \, x^{5} + 78 \, x^{4} + 40 \, x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{7} + 105 \, x^{6} - 3 \, x^{5} - 6 \, x^{4} - 4 \, x^{3} - 3 \, x^{2} - 3 \, x - 1\right )}}\right ) + 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}}{x^{3} + x + 1}\right ) - \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{4} + 20 \, x^{3} + x^{2} + 2 \, x + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{4} + 2 \, x^{3} + x^{2} + 2 \, x + 1}\right ) - 9 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
1/6*(2*sqrt(3)*(-4)^(1/3)*x^2*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(7*x^7 + 8* x^5 + 8*x^4 + x^3 + 2*x^2 + x)*(3*x^3 + x + 1)^(2/3) - 6*sqrt(3)*(-4)^(1/3 )*(55*x^8 + 20*x^6 + 20*x^5 + x^4 + 2*x^3 + x^2)*(3*x^3 + x + 1)^(1/3) + s qrt(3)*(433*x^9 + 255*x^7 + 255*x^6 + 39*x^5 + 78*x^4 + 40*x^3 + 3*x^2 + 3 *x + 1))/(323*x^9 + 105*x^7 + 105*x^6 - 3*x^5 - 6*x^4 - 4*x^3 - 3*x^2 - 3* x - 1)) + 2*(-4)^(1/3)*x^2*log((3*(-4)^(2/3)*(3*x^3 + x + 1)^(1/3)*x^2 - 6 *(3*x^3 + x + 1)^(2/3)*x - (-4)^(1/3)*(x^3 + x + 1))/(x^3 + x + 1)) - (-4) ^(1/3)*x^2*log(-(6*(-4)^(1/3)*(7*x^4 + x^2 + x)*(3*x^3 + x + 1)^(2/3) - (- 4)^(2/3)*(55*x^6 + 20*x^4 + 20*x^3 + x^2 + 2*x + 1) - 24*(4*x^5 + x^3 + x^ 2)*(3*x^3 + x + 1)^(1/3))/(x^6 + 2*x^4 + 2*x^3 + x^2 + 2*x + 1)) - 9*(3*x^ 3 + x + 1)^(2/3))/x^2
Timed out. \[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx=\int \frac {\left (2\,x+3\right )\,{\left (3\,x^3+x+1\right )}^{2/3}}{x^3\,\left (x^3+x+1\right )} \,d x \]