Integrand size = 28, antiderivative size = 111 \[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\frac {3 \left (2+x^3+x^5\right )^{2/3}}{4 x^2}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x^3+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{2+x^3+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{2+x^3+x^5}+\left (2+x^3+x^5\right )^{2/3}\right ) \]
3/4*(x^5+x^3+2)^(2/3)/x^2-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^5+x^3+2)^(1 /3)))+1/2*ln(-x+(x^5+x^3+2)^(1/3))-1/4*ln(x^2+x*(x^5+x^3+2)^(1/3)+(x^5+x^3 +2)^(2/3))
Time = 1.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\frac {1}{4} \left (\frac {3 \left (2+x^3+x^5\right )^{2/3}}{x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x^3+x^5}}\right )+2 \log \left (-x+\sqrt [3]{2+x^3+x^5}\right )-\log \left (x^2+x \sqrt [3]{2+x^3+x^5}+\left (2+x^3+x^5\right )^{2/3}\right )\right ) \]
((3*(2 + x^3 + x^5)^(2/3))/x^2 - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x^3 + x^5)^(1/3))] + 2*Log[-x + (2 + x^3 + x^5)^(1/3)] - Log[x^2 + x*(2 + x^3 + x^5)^(1/3) + (2 + x^3 + x^5)^(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.
| \(\displaystyle \int \frac {\left (x^5-3\right ) \left (x^5+x^3+2\right )^{2/3}}{x^3 \left (x^5+2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5 x^2 \left (x^5+x^3+2\right )^{2/3}}{2 \left (x^5+2\right )}-\frac {3 \left (x^5+x^3+2\right )^{2/3}}{2 x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \int \frac {\left (x^5+x^3+2\right )^{2/3}}{x^3}dx+\frac {\int \frac {\left (x^5+x^3+2\right )^{2/3}}{x+\sqrt [5]{2}}dx}{2\ 2^{2/5}}-\frac {(-1)^{3/5} \int \frac {\left (x^5+x^3+2\right )^{2/3}}{\sqrt [5]{2}-\sqrt [5]{-1} x}dx}{2\ 2^{2/5}}-\frac {\sqrt [5]{-1} \int \frac {\left (x^5+x^3+2\right )^{2/3}}{(-1)^{2/5} x+\sqrt [5]{2}}dx}{2\ 2^{2/5}}+\frac {(-1)^{4/5} \int \frac {\left (x^5+x^3+2\right )^{2/3}}{\sqrt [5]{2}-(-1)^{3/5} x}dx}{2\ 2^{2/5}}+\frac {1}{2} \left (-\frac {1}{2}\right )^{2/5} \int \frac {\left (x^5+x^3+2\right )^{2/3}}{(-1)^{4/5} x+\sqrt [5]{2}}dx\) |
3.17.44.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.63 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}}+\left (x^{5}+x^{3}+2\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{5}+x^{3}+2\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}}}{4 x^{2}}\) | \(110\) |
| risch | \(\frac {3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}}}{4 x^{2}}+\operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+x^{5}-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+2}{x^{5}+2}\right )-\frac {\ln \left (-\frac {2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}+4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+4}{x^{5}+2}\right )}{2}-\ln \left (-\frac {2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}+4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+4}{x^{5}+2}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\) | \(381\) |
| trager | \(\text {Expression too large to display}\) | \(633\) |
1/4*(2*3^(1/2)*arctan(1/3*(x+2*(x^5+x^3+2)^(1/3))*3^(1/2)/x)*x^2-ln((x^2+x *(x^5+x^3+2)^(1/3)+(x^5+x^3+2)^(2/3))/x^2)*x^2+2*ln((-x+(x^5+x^3+2)^(1/3)) /x)*x^2+3*(x^5+x^3+2)^(2/3))/x^2
Time = 4.89 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {240779826 \, \sqrt {3} {\left (x^{5} + x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 64389332 \, \sqrt {3} {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18550880 \, x^{5} + 88195247 \, x^{3} + 37101760\right )}}{3 \, {\left (2863288 \, x^{5} + 152584579 \, x^{3} + 5726576\right )}}\right ) - x^{2} \log \left (\frac {x^{5} + 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} x + 2}{x^{5} + 2}\right ) - 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}}}{4 \, x^{2}} \]
-1/4*(2*sqrt(3)*x^2*arctan(-1/3*(240779826*sqrt(3)*(x^5 + x^3 + 2)^(1/3)*x ^2 - 64389332*sqrt(3)*(x^5 + x^3 + 2)^(2/3)*x + sqrt(3)*(18550880*x^5 + 88 195247*x^3 + 37101760))/(2863288*x^5 + 152584579*x^3 + 5726576)) - x^2*log ((x^5 + 3*(x^5 + x^3 + 2)^(1/3)*x^2 - 3*(x^5 + x^3 + 2)^(2/3)*x + 2)/(x^5 + 2)) - 3*(x^5 + x^3 + 2)^(2/3))/x^2
Timed out. \[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} {\left (x^{5} - 3\right )}}{{\left (x^{5} + 2\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} {\left (x^{5} - 3\right )}}{{\left (x^{5} + 2\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx=\int \frac {\left (x^5-3\right )\,{\left (x^5+x^3+2\right )}^{2/3}}{x^3\,\left (x^5+2\right )} \,d x \]