Integrand size = 33, antiderivative size = 151 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
-3/2*(x^3-x^2+2)^(2/3)/x^2+2^(2/3)*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^ 3-x^2+2)^(1/3)))-2^(2/3)*ln(-2*x+2^(2/3)*(x^3-x^2+2)^(1/3))+1/2*ln(2*x^2+2 ^(2/3)*x*(x^3-x^2+2)^(1/3)+2^(1/3)*(x^3-x^2+2)^(2/3))*2^(2/3)
Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
(-3*(2 - x^2 + x^3)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(2 - x^2 + x^3)^(1/3))] - 2^(2/3)*Log[-2*x + 2^(2/3)*(2 - x^2 + x^3)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(2 - x^2 + x^3)^(1/3) + 2^(1/3)*(2 - x^2 + 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.
| \(\displaystyle \int \frac {\left (x^2-6\right ) \left (x^3-x^2+2\right )^{2/3}}{x^3 \left (x^3+x^2-2\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {(-x-3) \left (x^3-x^2+2\right )^{2/3} \left (x^2-6\right )}{5 x^3 \left (x^2+2 x+2\right )}+\frac {\left (x^3-x^2+2\right )^{2/3} \left (x^2-6\right )}{5 (x-1) x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {9 i \left (x^3-x^2+2\right )^{2/3} \text {Subst}\left (\int \frac {\left (x+\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3} \left (x^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )\right )^{2/3}}{-2 x-\left (\frac {8}{3}-2 i\right )}dx,x,x-\frac {1}{3}\right )}{\left (3 x+\sqrt [3]{26-15 \sqrt {3}}+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}-1\right )^{2/3} \left ((3 x-1)^2+\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (1-3 x)}{\sqrt [3]{26-15 \sqrt {3}}}+\left (26-15 \sqrt {3}\right )^{2/3}+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}-1\right )^{2/3}}-\frac {9 \left (x^3-x^2+2\right )^{2/3} \text {Subst}\left (\int \frac {\left (x+\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3} \left (x^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )\right )^{2/3}}{x-\frac {2}{3}}dx,x,x-\frac {1}{3}\right )}{\left (3 x+\sqrt [3]{26-15 \sqrt {3}}+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}-1\right )^{2/3} \left ((3 x-1)^2+\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (1-3 x)}{\sqrt [3]{26-15 \sqrt {3}}}+\left (26-15 \sqrt {3}\right )^{2/3}+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}-1\right )^{2/3}}+\frac {27 \left (x^3-x^2+2\right )^{2/3} \text {Subst}\left (\int \frac {\left (x+\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3} \left (x^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )\right )^{2/3}}{\left (x+\frac {1}{3}\right )^3}dx,x,x-\frac {1}{3}\right )}{\left (3 x+\sqrt [3]{26-15 \sqrt {3}}+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}-1\right )^{2/3} \left ((3 x-1)^2+\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (1-3 x)}{\sqrt [3]{26-15 \sqrt {3}}}+\left (26-15 \sqrt {3}\right )^{2/3}+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}-1\right )^{2/3}}+\frac {9 \left (x^3-x^2+2\right )^{2/3} \text {Subst}\left (\int \frac {\left (x+\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3} \left (x^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )\right )^{2/3}}{x+\frac {1}{3}}dx,x,x-\frac {1}{3}\right )}{\left (3 x+\sqrt [3]{26-15 \sqrt {3}}+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}-1\right )^{2/3} \left ((3 x-1)^2+\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (1-3 x)}{\sqrt [3]{26-15 \sqrt {3}}}+\left (26-15 \sqrt {3}\right )^{2/3}+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}-1\right )^{2/3}}+\frac {9 i \left (x^3-x^2+2\right )^{2/3} \text {Subst}\left (\int \frac {\left (x+\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3} \left (x^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )\right )^{2/3}}{2 x+\left (\frac {8}{3}+2 i\right )}dx,x,x-\frac {1}{3}\right )}{\left (3 x+\sqrt [3]{26-15 \sqrt {3}}+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}-1\right )^{2/3} \left ((3 x-1)^2+\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (1-3 x)}{\sqrt [3]{26-15 \sqrt {3}}}+\left (26-15 \sqrt {3}\right )^{2/3}+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}-1\right )^{2/3}}\) |
3.21.86.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 16.01 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} x +\left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}}{x}\right ) x^{2}-3 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(140\) |
| risch | \(\text {Expression too large to display}\) | \(735\) |
| trager | \(\text {Expression too large to display}\) | \(1526\) |
1/2*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3-x^2+2)^(1/3)) )*x^2+2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x^3-x^2+2)^(1/3)*x+(x^3-x^2+2)^(2/3 ))/x^2)*x^2-2*2^(2/3)*ln((-2^(1/3)*x+(x^3-x^2+2)^(1/3))/x)*x^2-3*(x^3-x^2+ 2)^(2/3))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (122) = 244\).
Time = 8.45 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.78 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{6} - x^{5} - 8 \, x^{4} + 4 \, x^{3} - 4 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{7} + x^{6} + 32 \, x^{5} - 4 \, x^{4} + 4 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{8} + 33 \, x^{7} + 221 \, x^{6} - 132 \, x^{5} + 6 \, x^{4} + 132 \, x^{3} - 12 \, x^{2} + 8\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{8} + 3 \, x^{7} + 211 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} - 8\right )}}\right ) - 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x^{2} - 2\right )}}{x^{3} + x^{2} - 2}\right ) + \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x^{3} + 2 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{5} + x^{4} + 32 \, x^{3} - 4 \, x^{2} + 4\right )} - 24 \, {\left (2 \, x^{5} - x^{4} + 2 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{5} + x^{4} - 4 \, x^{3} - 4 \, x^{2} + 4}\right ) + 9 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
-1/6*(2*sqrt(3)*(-4)^(1/3)*x^2*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 + 4 *x^6 - x^5 - 8*x^4 + 4*x^3 - 4*x)*(x^3 - x^2 + 2)^(2/3) + 6*sqrt(3)*(-4)^( 1/3)*(19*x^8 - 16*x^7 + x^6 + 32*x^5 - 4*x^4 + 4*x^2)*(x^3 - x^2 + 2)^(1/3 ) - sqrt(3)*(71*x^9 - 111*x^8 + 33*x^7 + 221*x^6 - 132*x^5 + 6*x^4 + 132*x ^3 - 12*x^2 + 8))/(109*x^9 - 105*x^8 + 3*x^7 + 211*x^6 - 12*x^5 - 6*x^4 + 12*x^3 + 12*x^2 - 8)) - 2*(-4)^(1/3)*x^2*log(-(3*(-4)^(2/3)*(x^3 - x^2 + 2 )^(1/3)*x^2 - 6*(x^3 - x^2 + 2)^(2/3)*x + (-4)^(1/3)*(x^3 + x^2 - 2))/(x^3 + x^2 - 2)) + (-4)^(1/3)*x^2*log(-(6*(-4)^(1/3)*(5*x^4 - x^3 + 2*x)*(x^3 - x^2 + 2)^(2/3) - (-4)^(2/3)*(19*x^6 - 16*x^5 + x^4 + 32*x^3 - 4*x^2 + 4) - 24*(2*x^5 - x^4 + 2*x^2)*(x^3 - x^2 + 2)^(1/3))/(x^6 + 2*x^5 + x^4 - 4* x^3 - 4*x^2 + 4)) + 9*(x^3 - x^2 + 2)^(2/3))/x^2
\[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - 2 x + 2\right )\right )^{\frac {2}{3}} \left (x^{2} - 6\right )}{x^{3} \left (x - 1\right ) \left (x^{2} + 2 x + 2\right )}\, dx \]
\[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=\int \frac {\left (x^2-6\right )\,{\left (x^3-x^2+2\right )}^{2/3}}{x^3\,\left (x^3+x^2-2\right )} \,d x \]