Integrand size = 28, antiderivative size = 111 \[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\frac {3 \sqrt [3]{-2+x^3+x^5}}{2 x}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^3+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-2+x^3+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{-2+x^3+x^5}+\left (-2+x^3+x^5\right )^{2/3}\right ) \]
3/2*(x^5+x^3-2)^(1/3)/x+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 = 1.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\frac {3 \sqrt [3]{-2+x^3+x^5}}{2 x}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^3+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-2+x^3+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{-2+x^3+x^5}+\left (-2+x^3+x^5\right )^{2/3}\right ) \]
(3*(-2 + x^3 + x^5)^(1/3))/(2*x) + (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)]/2 - 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.
| \(\displaystyle \int \frac {\left (x^5+3\right ) \sqrt [3]{x^5+x^3-2}}{x^2 \left (x^5-2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5 x^3 \sqrt [3]{x^5+x^3-2}}{2 \left (x^5-2\right )}-\frac {3 \sqrt [3]{x^5+x^3-2}}{2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \frac {\sqrt [3]{x^5+x^3-2}}{\sqrt [5]{2}-x}dx}{2 \sqrt [5]{2}}-\frac {(-1)^{2/5} \int \frac {\sqrt [3]{x^5+x^3-2}}{\sqrt [5]{-1} x+\sqrt [5]{2}}dx}{2 \sqrt [5]{2}}-\frac {(-1)^{4/5} \int \frac {\sqrt [3]{x^5+x^3-2}}{\sqrt [5]{2}-(-1)^{2/5} x}dx}{2 \sqrt [5]{2}}+\frac {1}{2} \sqrt [5]{-\frac {1}{2}} \int \frac {\sqrt [3]{x^5+x^3-2}}{(-1)^{3/5} x+\sqrt [5]{2}}dx+\frac {(-1)^{3/5} \int \frac {\sqrt [3]{x^5+x^3-2}}{\sqrt [5]{2}-(-1)^{4/5} x}dx}{2 \sqrt [5]{2}}-\frac {3}{2} \int \frac {\sqrt [3]{x^5+x^3-2}}{x^2}dx\) |
3.17.43.3.1 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 = 6.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x -\ln \left (\frac {x^{2}+x \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}+\left (x^{5}+x^{3}-2\right )^{\frac {2}{3}}}{x^{2}}\right ) x +2 \ln \left (\frac {-x +\left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}}{x}\right ) x +6 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}}{4 x}\) | \(104\) |
| trager | \(\frac {3 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}}{2 x}+\frac {\ln \left (\frac {-3687964213260096 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}-1951741557610164 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}+13829865799725360 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-88450351904093 x^{5}+2398485078249264 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x +136263463016940 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-1382259724622424 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-11355288584745 \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x +211229045105517 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} x^{2}-103833021800457 x^{3}+7375928426520192 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3903483115220328 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+176900703808186}{x^{5}-2}\right )}{2}+6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {-1472859748183680 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}+767599022412408 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}+5523224055688800 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-73067682007729 x^{5}+2398485078249264 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x -2534748541266204 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+596532134324340 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+211229045105517 \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x -11355288584745 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} x^{2}-161518033911822 x^{3}+2945719496367360 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-1535198044824816 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+146135364015458}{x^{5}-2}\right )\) | \(423\) |
| risch | \(\text {Expression too large to display}\) | \(1123\) |
1/4*(-2*3^(1/2)*arctan(1/3*(x+2*(x^5+x^3-2)^(1/3))*3^(1/2)/x)*x-ln((x^2+x* (x^5+x^3-2)^(1/3)+(x^5+x^3-2)^(2/3))/x^2)*x+2*ln((-x+(x^5+x^3-2)^(1/3))/x) *x+6*(x^5+x^3-2)^(1/3))/x
Time = 3.65 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.23 \[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {240779826 \, \sqrt {3} {\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}} x^{2} - 64389332 \, \sqrt {3} {\left (x^{5} + x^{3} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18550880 \, x^{5} + 88195247 \, x^{3} - 37101760\right )}}{3 \, {\left (2863288 \, x^{5} + 152584579 \, x^{3} - 5726576\right )}}\right ) + x \log \left (\frac {x^{5} + 3 \, {\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} + x^{3} - 2\right )}^{\frac {2}{3}} x - 2}{x^{5} - 2}\right ) + 6 \, {\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}}}{4 \, x} \]
1/4*(2*sqrt(3)*x*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 + 88195 247*x^3 - 37101760))/(2863288*x^5 + 152584579*x^3 - 5726576)) + x*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)) + 6*(x^5 + x^3 - 2)^(1/3))/x
\[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right ) \left (x^{4} + x^{3} + 2 x^{2} + 2 x + 2\right )} \left (x^{5} + 3\right )}{x^{2} \left (x^{5} - 2\right )}\, dx \]
\[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}} {\left (x^{5} + 3\right )}}{{\left (x^{5} - 2\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}} {\left (x^{5} + 3\right )}}{{\left (x^{5} - 2\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \left (-2+x^5\right )} \, dx=\int \frac {\left (x^5+3\right )\,{\left (x^5+x^3-2\right )}^{1/3}}{x^2\,\left (x^5-2\right )} \,d x \]