Integrand size = 31, antiderivative size = 175 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \]
3/4*(x^3+2*x-1)^(2/3)/x^2-3/4*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(1/3 )*(x^3+2*x-1)^(1/3)))*2^(1/3)+1/4*2^(1/3)*3^(2/3)*ln(-3*x+2^(1/3)*3^(2/3)* (x^3+2*x-1)^(1/3))-1/8*2^(1/3)*3^(2/3)*ln(3*x^2+2^(1/3)*3^(2/3)*x*(x^3+2*x -1)^(1/3)+2^(2/3)*3^(1/3)*(x^3+2*x-1)^(2/3))
Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \]
(3*(-1 + 2*x + x^3)^(2/3))/(4*x^2) - (3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3 )*x + 2*2^(1/3)*(-1 + 2*x + x^3)^(1/3))])/(2*2^(2/3)) + ((3/2)^(2/3)*Log[- 3*x + 2^(1/3)*3^(2/3)*(-1 + 2*x + x^3)^(1/3)])/2 - ((3/2)^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + 2*x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + 2*x + x ^3)^(2/3)])/4
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.
| \(\displaystyle \int \frac {(4 x-3) \left (x^3+2 x-1\right )^{2/3}}{x^3 \left (x^3-4 x+2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 \left (x^3+2 x-1\right )^{2/3}}{x}-\frac {3 \left (x^3+2 x-1\right )^{2/3}}{2 x^3}+\frac {\left (x^3+2 x-1\right )^{2/3} \left (4 x^2+2 x-13\right )}{2 \left (x^3-4 x+2\right )}-\frac {\left (x^3+2 x-1\right )^{2/3}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {27 \sqrt [3]{2} \left (x^3+2 x-1\right )^{2/3} \int \frac {\left (x+\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}\right )^{2/3} \left (x^2-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )\right )^{2/3}}{x^3}dx}{\left (6 x+\sqrt [3]{6} \left (4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}\right )\right )^{2/3} \left (18 x^2-6 \sqrt [3]{3} \left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+12\right )^{2/3}}-\frac {18 \sqrt [3]{2} \left (x^3+2 x-1\right )^{2/3} \int \frac {\left (x+\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}\right )^{2/3} \left (x^2-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )\right )^{2/3}}{x^2}dx}{\left (6 x+\sqrt [3]{6} \left (4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}\right )\right )^{2/3} \left (18 x^2-6 \sqrt [3]{3} \left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+12\right )^{2/3}}-\frac {36 \sqrt [3]{2} \left (x^3+2 x-1\right )^{2/3} \int \frac {\left (x+\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}\right )^{2/3} \left (x^2-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )\right )^{2/3}}{x}dx}{\left (6 x+\sqrt [3]{6} \left (4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}\right )\right )^{2/3} \left (18 x^2-6 \sqrt [3]{3} \left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+12\right )^{2/3}}-\frac {13}{2} \int \frac {\left (x^3+2 x-1\right )^{2/3}}{x^3-4 x+2}dx+\int \frac {x \left (x^3+2 x-1\right )^{2/3}}{x^3-4 x+2}dx+2 \int \frac {x^2 \left (x^3+2 x-1\right )^{2/3}}{x^3-4 x+2}dx\) |
3.23.94.3.1 Defintions of rubi rules used
Time = 16.96 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 \left (x^{3}+2 x -1\right )^{\frac {1}{3}}}{x}\right )-2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}} x +2 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{x^{2}}\right )-2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (2\right )+6 \,3^{\frac {1}{6}} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{2}+6 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{8 x^{2}}\) | \(169\) |
| trager | \(\text {Expression too large to display}\) | \(986\) |
| risch | \(\text {Expression too large to display}\) | \(1034\) |
1/8*(2*2^(1/3)*3^(2/3)*x^2*ln((-2^(2/3)*3^(1/3)*x+2*(x^3+2*x-1)^(1/3))/x)- 2^(1/3)*3^(2/3)*x^2*ln((2^(1/3)*3^(2/3)*x^2+2^(2/3)*3^(1/3)*(x^3+2*x-1)^(1 /3)*x+2*(x^3+2*x-1)^(2/3))/x^2)-2^(1/3)*3^(2/3)*x^2*ln(2)+6*3^(1/6)*2^(1/3 )*arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x^3+2*x-1)^(1/3)+3*x)/x)*x^2+6*(x ^3+2*x-1)^(2/3))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (133) = 266\).
Time = 10.50 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.44 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=-\frac {4 \cdot 9^{\frac {1}{3}} 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 9^{\frac {2}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{7} - 14 \, x^{5} + 7 \, x^{4} - 8 \, x^{3} + 8 \, x^{2} - 2 \, x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} - 12 \cdot 9^{\frac {1}{3}} {\left (55 \, x^{8} + 100 \, x^{6} - 50 \, x^{5} + 16 \, x^{4} - 16 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} + 1200 \, x^{7} - 600 \, x^{6} + 816 \, x^{5} - 816 \, x^{4} + 268 \, x^{3} - 96 \, x^{2} + 48 \, x - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} + 960 \, x^{7} - 480 \, x^{6} + 48 \, x^{5} - 48 \, x^{4} - 52 \, x^{3} + 96 \, x^{2} - 48 \, x + 8\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} x^{2} - 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (x^{3} - 4 \, x + 2\right )} - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} x}{x^{3} - 4 \, x + 2}\right ) + 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (\frac {18 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{4} + 2 \, x^{2} - x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (55 \, x^{6} + 100 \, x^{4} - 50 \, x^{3} + 16 \, x^{2} - 16 \, x + 4\right )} + 54 \, {\left (7 \, x^{5} + 8 \, x^{3} - 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}}}{x^{6} - 8 \, x^{4} + 4 \, x^{3} + 16 \, x^{2} - 16 \, x + 4}\right ) - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \]
-1/48*(4*9^(1/3)*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(4*9^(2/3) *4^(2/3)*(4*x^7 - 14*x^5 + 7*x^4 - 8*x^3 + 8*x^2 - 2*x)*(x^3 + 2*x - 1)^(2 /3) - 12*9^(1/3)*(55*x^8 + 100*x^6 - 50*x^5 + 16*x^4 - 16*x^3 + 4*x^2)*(x^ 3 + 2*x - 1)^(1/3) - 4^(1/3)*(377*x^9 + 1200*x^7 - 600*x^6 + 816*x^5 - 816 *x^4 + 268*x^3 - 96*x^2 + 48*x - 8))/(487*x^9 + 960*x^7 - 480*x^6 + 48*x^5 - 48*x^4 - 52*x^3 + 96*x^2 - 48*x + 8)) - 2*9^(1/3)*4^(2/3)*x^2*log(-(6*9 ^(2/3)*4^(1/3)*(x^3 + 2*x - 1)^(1/3)*x^2 - 9^(1/3)*4^(2/3)*(x^3 - 4*x + 2) - 36*(x^3 + 2*x - 1)^(2/3)*x)/(x^3 - 4*x + 2)) + 9^(1/3)*4^(2/3)*x^2*log( (18*9^(1/3)*4^(2/3)*(4*x^4 + 2*x^2 - x)*(x^3 + 2*x - 1)^(2/3) + 9^(2/3)*4^ (1/3)*(55*x^6 + 100*x^4 - 50*x^3 + 16*x^2 - 16*x + 4) + 54*(7*x^5 + 8*x^3 - 4*x^2)*(x^3 + 2*x - 1)^(1/3))/(x^6 - 8*x^4 + 4*x^3 + 16*x^2 - 16*x + 4)) - 36*(x^3 + 2*x - 1)^(2/3))/x^2
\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int \frac {\left (4 x - 3\right ) \left (x^{3} + 2 x - 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} - 4 x + 2\right )}\, dx \]
\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} {\left (4 \, x - 3\right )}}{{\left (x^{3} - 4 \, x + 2\right )} x^{3}} \,d x } \]
\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} {\left (4 \, x - 3\right )}}{{\left (x^{3} - 4 \, x + 2\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int \frac {\left (4\,x-3\right )\,{\left (x^3+2\,x-1\right )}^{2/3}}{x^3\,\left (x^3-4\,x+2\right )} \,d x \]