Integrand size = 41, antiderivative size = 154 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {4 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{5 x^5}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}} \] Output:
4/5*(3*x^4+2)^(1/4)*(6*x^4-1)/x^5+1/2*arctan(x/(3*x^4+2)^(1/4))+5/4*arctan (2^(1/2)*x*(3*x^4+2)^(1/4)/(-x^2+(3*x^4+2)^(1/2)))*2^(1/2)-1/2*arctanh(x/( 3*x^4+2)^(1/4))-5/4*arctanh(2^(1/2)*x*(3*x^4+2)^(1/4)/(x^2+(3*x^4+2)^(1/2) ))*2^(1/2)
Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {1}{20} \left (\frac {16 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{x^5}+10 \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+25 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )-10 \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-25 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )\right ) \] Input:
Integrate[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4) ),x]
Output:
((16*(2 + 3*x^4)^(1/4)*(-1 + 6*x^4))/x^5 + 10*ArcTan[x/(2 + 3*x^4)^(1/4)] + 25*Sqrt[2]*ArcTan[(Sqrt[2]*x*(2 + 3*x^4)^(1/4))/(-x^2 + Sqrt[2 + 3*x^4]) ] - 10*ArcTanh[x/(2 + 3*x^4)^(1/4)] - 25*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(2 + 3 *x^4)^(1/4))/(x^2 + Sqrt[2 + 3*x^4])])/20
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.93 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {\sqrt [4]{3 x^4+2} \left (x^8+6 x^4+4\right )}{x^6 \left (x^4+1\right ) \left (2 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {4 \sqrt [4]{3 x^4+2}}{x^6}+\frac {\sqrt [4]{3 x^4+2} x^2}{x^4+1}+\frac {10 \sqrt [4]{3 x^4+2} x^2}{2 x^4+1}-\frac {6 \sqrt [4]{3 x^4+2}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-3 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )+\frac {6 \sqrt [4]{3 x^4+2}}{x}-\frac {2 \left (3 x^4+2\right )^{5/4}}{5 x^5}\) |
Input:
Int[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4)),x]
Output:
(6*(2 + 3*x^4)^(1/4))/x - (2*(2 + 3*x^4)^(5/4))/(5*x^5) + (10*2^(1/4)*x^3* AppellF1[3/4, 1, -1/4, 7/4, -2*x^4, (-3*x^4)/2])/3 + (2^(1/4)*x^3*AppellF1 [3/4, 1, -1/4, 7/4, -x^4, (-3*x^4)/2])/3 + 3*3^(1/4)*ArcTan[(3^(1/4)*x)/(2 + 3*x^4)^(1/4)] - 3*3^(1/4)*ArcTanh[(3^(1/4)*x)/(2 + 3*x^4)^(1/4)]
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 = 13.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.43
| method | result | size |
| pseudoelliptic | \(\frac {-25 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {3 x^{4}+2}}{-\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {3 x^{4}+2}}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x^{5}-10 \ln \left (\frac {x +\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}+10 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}-20 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}+192 \left (3 x^{4}+2\right )^{\frac {1}{4}} x^{4}-32 \left (3 x^{4}+2\right )^{\frac {1}{4}}}{40 x^{5}}\) | \(220\) |
| trager | \(\frac {4 \left (3 x^{4}+2\right )^{\frac {1}{4}} \left (6 x^{4}-1\right )}{5 x^{5}}-\frac {\ln \left (-\frac {\left (3 x^{4}+2\right )^{\frac {3}{4}} x +x^{2} \sqrt {3 x^{4}+2}+\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}+2 x^{4}+1}{x^{4}+1}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+1}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {3 x^{4}+2}\, x^{2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2 x^{4}+1}\right )}{4}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}-\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\left (3 x^{4}+2\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2 x^{4}+1}\right )}{4}\) | \(400\) |
| risch | \(\text {Expression too large to display}\) | \(987\) |
Input:
int((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x,method=_RETURNVE RBOSE)
Output:
1/40*(-25*ln(((3*x^4+2)^(1/4)*x*2^(1/2)+x^2+(3*x^4+2)^(1/2))/(-(3*x^4+2)^( 1/4)*x*2^(1/2)+x^2+(3*x^4+2)^(1/2)))*2^(1/2)*x^5-50*arctan(((3*x^4+2)^(1/4 )*2^(1/2)+x)/x)*2^(1/2)*x^5-50*arctan(((3*x^4+2)^(1/4)*2^(1/2)-x)/x)*2^(1/ 2)*x^5-10*ln((x+(3*x^4+2)^(1/4))/x)*x^5+10*ln(((3*x^4+2)^(1/4)-x)/x)*x^5-2 0*arctan(1/x*(3*x^4+2)^(1/4))*x^5+192*(3*x^4+2)^(1/4)*x^4-32*(3*x^4+2)^(1/ 4))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (123) = 246\).
Time = 3.48 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.97 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=-\frac {50 \, \sqrt {2} x^{5} \arctan \left (\frac {4 \, x^{8} + 4 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) + 50 \, \sqrt {2} x^{5} \arctan \left (-\frac {4 \, x^{8} + 4 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) + 25 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) - 25 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) - 20 \, x^{5} \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 20 \, x^{5} \log \left (-\frac {2 \, x^{4} - {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {3 \, x^{4} + 2} x^{2} - {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}\right ) - 64 \, {\left (6 \, x^{4} - 1\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{80 \, x^{5}} \] Input:
integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm ="fricas")
Output:
-1/80*(50*sqrt(2)*x^5*arctan((4*x^8 + 4*x^4 + sqrt(2)*(3*x^4 + 2)^(3/4)*x + sqrt(2)*(4*x^7 + 3*x^3)*(3*x^4 + 2)^(1/4) + 2*(2*x^6 + x^2)*sqrt(3*x^4 + 2) + 1)/(8*x^8 + 4*x^4 - 1)) + 50*sqrt(2)*x^5*arctan(-(4*x^8 + 4*x^4 - sq rt(2)*(3*x^4 + 2)^(3/4)*x - sqrt(2)*(4*x^7 + 3*x^3)*(3*x^4 + 2)^(1/4) + 2* (2*x^6 + x^2)*sqrt(3*x^4 + 2) + 1)/(8*x^8 + 4*x^4 - 1)) + 25*sqrt(2)*x^5*l og((2*x^4 + sqrt(2)*(3*x^4 + 2)^(1/4)*x^3 + 2*sqrt(3*x^4 + 2)*x^2 + sqrt(2 )*(3*x^4 + 2)^(3/4)*x + 1)/(2*x^4 + 1)) - 25*sqrt(2)*x^5*log((2*x^4 - sqrt (2)*(3*x^4 + 2)^(1/4)*x^3 + 2*sqrt(3*x^4 + 2)*x^2 - sqrt(2)*(3*x^4 + 2)^(3 /4)*x + 1)/(2*x^4 + 1)) - 20*x^5*arctan(((3*x^4 + 2)^(1/4)*x^3 + (3*x^4 + 2)^(3/4)*x)/(x^4 + 1)) - 20*x^5*log(-(2*x^4 - (3*x^4 + 2)^(1/4)*x^3 + sqrt (3*x^4 + 2)*x^2 - (3*x^4 + 2)^(3/4)*x + 1)/(x^4 + 1)) - 64*(6*x^4 - 1)*(3* x^4 + 2)^(1/4))/x^5
\[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int \frac {\sqrt [4]{3 x^{4} + 2} \left (x^{8} + 6 x^{4} + 4\right )}{x^{6} \left (x^{4} + 1\right ) \left (2 x^{4} + 1\right )}\, dx \] Input:
integrate((3*x**4+2)**(1/4)*(x**8+6*x**4+4)/x**6/(x**4+1)/(2*x**4+1),x)
Output:
Integral((3*x**4 + 2)**(1/4)*(x**8 + 6*x**4 + 4)/(x**6*(x**4 + 1)*(2*x**4 + 1)), x)
\[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int { \frac {{\left (x^{8} + 6 \, x^{4} + 4\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )} x^{6}} \,d x } \] Input:
integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm ="maxima")
Output:
integrate((x^8 + 6*x^4 + 4)*(3*x^4 + 2)^(1/4)/((2*x^4 + 1)*(x^4 + 1)*x^6), x)
Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=-\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) + \frac {5}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (\frac {2}{x^{4}} + 3\right )}}{5 \, x} + \frac {6 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - 1\right ) \] Input:
integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm ="giac")
Output:
-5/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^4 + 2)^(1/4)/x)) - 5/4*s qrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(3*x^4 + 2)^(1/4)/x)) - 5/8*sqrt(2 )*log(sqrt(2)*(3*x^4 + 2)^(1/4)/x + sqrt(3*x^4 + 2)/x^2 + 1) + 5/8*sqrt(2) *log(-sqrt(2)*(3*x^4 + 2)^(1/4)/x + sqrt(3*x^4 + 2)/x^2 + 1) - 2/5*(3*x^4 + 2)^(1/4)*(2/x^4 + 3)/x + 6*(3*x^4 + 2)^(1/4)/x - 1/2*arctan((3*x^4 + 2)^ (1/4)/x) - 1/4*log((3*x^4 + 2)^(1/4)/x + 1) + 1/4*log((3*x^4 + 2)^(1/4)/x - 1)
Timed out. \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int \frac {{\left (3\,x^4+2\right )}^{1/4}\,\left (x^8+6\,x^4+4\right )}{x^6\,\left (x^4+1\right )\,\left (2\,x^4+1\right )} \,d x \] Input:
int(((3*x^4 + 2)^(1/4)*(6*x^4 + x^8 + 4))/(x^6*(x^4 + 1)*(2*x^4 + 1)),x)
Output:
int(((3*x^4 + 2)^(1/4)*(6*x^4 + x^8 + 4))/(x^6*(x^4 + 1)*(2*x^4 + 1)), x)
\[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {253 \left (3 x^{4}+2\right )^{\frac {1}{4}} x^{4}-48 \left (3 x^{4}+2\right )^{\frac {1}{4}}-70 \left (\int \frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}}{6 x^{14}+13 x^{10}+9 x^{6}+2 x^{2}}d x \right ) x^{5}+40 \left (\int \frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{6}}{6 x^{12}+13 x^{8}+9 x^{4}+2}d x \right ) x^{5}+30 \left (\int \frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{2}}{6 x^{12}+13 x^{8}+9 x^{4}+2}d x \right ) x^{5}}{60 x^{5}} \] Input:
int((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x)
Output:
(253*(3*x**4 + 2)**(1/4)*x**4 - 48*(3*x**4 + 2)**(1/4) - 70*int((3*x**4 + 2)**(1/4)/(6*x**14 + 13*x**10 + 9*x**6 + 2*x**2),x)*x**5 + 40*int(((3*x**4 + 2)**(1/4)*x**6)/(6*x**12 + 13*x**8 + 9*x**4 + 2),x)*x**5 + 30*int(((3*x **4 + 2)**(1/4)*x**2)/(6*x**12 + 13*x**8 + 9*x**4 + 2),x)*x**5)/(60*x**5)