Integrand size = 45, antiderivative size = 85 \[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\frac {4 \sqrt [4]{-2+x^4+2 x^5} \left (10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right )}{45 x^9}+2 \arctan \left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right ) \]
4/45*(2*x^5+x^4-2)^(1/4)*(10*x^10+x^9+43*x^8-20*x^5-x^4+10)/x^9+2*arctan(x /(2*x^5+x^4-2)^(1/4))-2*arctanh(x/(2*x^5+x^4-2)^(1/4))
Time = 1.39 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\frac {4 \sqrt [4]{-2+x^4+2 x^5} \left (10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right )}{45 x^9}+2 \arctan \left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right ) \]
Integrate[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/ (x^10*(-1 + x^5)),x]
(4*(-2 + x^4 + 2*x^5)^(1/4)*(10 - x^4 - 20*x^5 + 43*x^8 + x^9 + 10*x^10))/ (45*x^9) + 2*ArcTan[x/(-2 + x^4 + 2*x^5)^(1/4)] - 2*ArcTanh[x/(-2 + x^4 + 2*x^5)^(1/4)]
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.
\(\displaystyle \int \frac {\left (x^5+4\right ) \sqrt [4]{2 x^5+x^4-2} \left (2 x^{10}+x^8-4 x^5+2\right )}{x^{10} \left (x^5-1\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt [4]{2 x^5+x^4-2}}{x-1}+\frac {6 \sqrt [4]{2 x^5+x^4-2}}{x^5}+2 \sqrt [4]{2 x^5+x^4-2}-\frac {8 \sqrt [4]{2 x^5+x^4-2}}{x^{10}}-\frac {4 \sqrt [4]{2 x^5+x^4-2}}{x^2}+\frac {\sqrt [4]{2 x^5+x^4-2} \left (-x^3+3 x^2+2 x+1\right )}{x^4+x^3+x^2+x+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \sqrt [4]{2 x^5+x^4-2}dx+\int \frac {\sqrt [4]{2 x^5+x^4-2}}{x-1}dx+6 \int \frac {\sqrt [4]{2 x^5+x^4-2}}{x^5}dx-8 \int \frac {\sqrt [4]{2 x^5+x^4-2}}{x^{10}}dx-4 \int \frac {\sqrt [4]{2 x^5+x^4-2}}{x^2}dx+\int \frac {\sqrt [4]{2 x^5+x^4-2}}{x^4+x^3+x^2+x+1}dx+2 \int \frac {x \sqrt [4]{2 x^5+x^4-2}}{x^4+x^3+x^2+x+1}dx+3 \int \frac {x^2 \sqrt [4]{2 x^5+x^4-2}}{x^4+x^3+x^2+x+1}dx-\int \frac {x^3 \sqrt [4]{2 x^5+x^4-2}}{x^4+x^3+x^2+x+1}dx\) |
3.12.51.3.1 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 = 10.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39
method | result | size |
pseudoelliptic | \(\frac {9 \ln \left (\frac {\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}-x}{x}\right ) x^{9}-9 \ln \left (\frac {\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}+x}{x}\right ) x^{9}-18 \arctan \left (\frac {\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}}{x}\right ) x^{9}+8 \left (x^{10}+\frac {1}{10} x^{9}+\frac {43}{10} x^{8}-2 x^{5}-\frac {1}{10} x^{4}+1\right ) \left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}}{9 x^{9}}\) | \(118\) |
trager | \(\frac {4 \left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} \left (10 x^{10}+x^{9}+43 x^{8}-20 x^{5}-x^{4}+10\right )}{45 x^{9}}+\ln \left (-\frac {-x^{5}+\left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}} x -\sqrt {2 x^{5}+x^{4}-2}\, x^{2}+\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} x^{3}-x^{4}+1}{\left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}-\sqrt {2 x^{5}+x^{4}-2}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}} x -\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )\) | \(237\) |
risch | \(\text {Expression too large to display}\) | \(1334\) |
1/9*(9*ln(((2*x^5+x^4-2)^(1/4)-x)/x)*x^9-9*ln(((2*x^5+x^4-2)^(1/4)+x)/x)*x ^9-18*arctan(1/x*(2*x^5+x^4-2)^(1/4))*x^9+8*(x^10+1/10*x^9+43/10*x^8-2*x^5 -1/10*x^4+1)*(2*x^5+x^4-2)^(1/4))/x^9
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (77) = 154\).
Time = 31.63 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.89 \[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\frac {45 \, x^{9} \arctan \left (\frac {{\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {3}{4}} x}{x^{5} - 1}\right ) + 45 \, x^{9} \log \left (-\frac {x^{5} + x^{4} - {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {2 \, x^{5} + x^{4} - 2} x^{2} - {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {3}{4}} x - 1}{x^{5} - 1}\right ) + 4 \, {\left (10 \, x^{10} + x^{9} + 43 \, x^{8} - 20 \, x^{5} - x^{4} + 10\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}}}{45 \, x^{9}} \]
integrate((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x, algorithm="fricas")
1/45*(45*x^9*arctan(((2*x^5 + x^4 - 2)^(1/4)*x^3 + (2*x^5 + x^4 - 2)^(3/4) *x)/(x^5 - 1)) + 45*x^9*log(-(x^5 + x^4 - (2*x^5 + x^4 - 2)^(1/4)*x^3 + sq rt(2*x^5 + x^4 - 2)*x^2 - (2*x^5 + x^4 - 2)^(3/4)*x - 1)/(x^5 - 1)) + 4*(1 0*x^10 + x^9 + 43*x^8 - 20*x^5 - x^4 + 10)*(2*x^5 + x^4 - 2)^(1/4))/x^9
Timed out. \[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{10} + x^{8} - 4 \, x^{5} + 2\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )} x^{10}} \,d x } \]
integrate((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x, algorithm="maxima")
\[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{10} + x^{8} - 4 \, x^{5} + 2\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )} x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx=\int \frac {\left (x^5+4\right )\,{\left (2\,x^5+x^4-2\right )}^{1/4}\,\left (2\,x^{10}+x^8-4\,x^5+2\right )}{x^{10}\,\left (x^5-1\right )} \,d x \]