Integrand size = 36, antiderivative size = 309 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=-\frac {3917 x \left (2+3 x^2\right )}{1280 \sqrt {2+5 x^2+3 x^4}}-\frac {13}{16} x \sqrt {2+5 x^2+3 x^4}+\frac {3}{40} x^3 \sqrt {2+5 x^2+3 x^4}+\frac {31 x \sqrt {2+5 x^2+3 x^4}}{64 \left (1+2 x^2\right )^2}+\frac {649 x \sqrt {2+5 x^2+3 x^4}}{128 \left (1+2 x^2\right )}+\frac {3917 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{640 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {1783 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{128 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {2617 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{128 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
-3917/1280*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-13/16*x*(3*x^4+5*x^2+2)^(1/2) +3/40*x^3*(3*x^4+5*x^2+2)^(1/2)+31/64*x*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2+ 649*x*(3*x^4+5*x^2+2)^(1/2)/(256*x^2+128)+3917/1280*2^(1/2)*(x^2+1)*((3*x^ 2+2)/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+ 2)^(1/2)-1783/256*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiA M(arctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+2617/384*(x^2+1)*Elliptic Pi(x*6^(1/2)/(6*x^2+4)^(1/2),-1/3,1/3*3^(1/2))*3^(1/2)/((x^2+1)/(3*x^2+2)) ^(1/2)/(3*x^4+5*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.65 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.66 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\frac {\frac {12 x \left (6070+24091 x^2+27619 x^4+4318 x^6-4704 x^8+576 x^{10}\right )}{\left (1+2 x^2\right )^2}+23502 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-27097 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-13085 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{7680 \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2)^3,x]
Output:
((12*x*(6070 + 24091*x^2 + 27619*x^4 + 4318*x^6 - 4704*x^8 + 576*x^10))/(1 + 2*x^2)^2 + (23502*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I* ArcSinh[Sqrt[3/2]*x], 2/3] - (27097*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^ 2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (13085*I)*Sqrt[3]*Sqrt[1 + x^2 ]*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(7680*Sqrt [2 + 5*x^2 + 3*x^4])
Time = 0.99 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2258, 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.
| \(\displaystyle \int \frac {\left (x^4-7 x^2+4\right ) \left (3 x^4+5 x^2+2\right )^{3/2}}{\left (2 x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (-\frac {93 x^4}{16 \sqrt {3 x^4+5 x^2+2}}-\frac {37 x^2}{4 \sqrt {3 x^4+5 x^2+2}}+\frac {555}{64 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {29}{8 \left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {31}{64 \left (2 x^2+1\right )^3 \sqrt {3 x^4+5 x^2+2}}+\frac {103}{32 \sqrt {3 x^4+5 x^2+2}}+\frac {9 x^6}{8 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1783 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{128 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {3917 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{640 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {69 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{16 \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {961 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{128 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {649 \sqrt {3 x^4+5 x^2+2} x}{128 \left (2 x^2+1\right )}+\frac {31 \sqrt {3 x^4+5 x^2+2} x}{64 \left (2 x^2+1\right )^2}-\frac {13}{16} \sqrt {3 x^4+5 x^2+2} x-\frac {3917 \left (3 x^2+2\right ) x}{1280 \sqrt {3 x^4+5 x^2+2}}+\frac {3}{40} \sqrt {3 x^4+5 x^2+2} x^3\) |
Input:
Int[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2)^3,x]
Output:
(-3917*x*(2 + 3*x^2))/(1280*Sqrt[2 + 5*x^2 + 3*x^4]) - (13*x*Sqrt[2 + 5*x^ 2 + 3*x^4])/16 + (3*x^3*Sqrt[2 + 5*x^2 + 3*x^4])/40 + (31*x*Sqrt[2 + 5*x^2 + 3*x^4])/(64*(1 + 2*x^2)^2) + (649*x*Sqrt[2 + 5*x^2 + 3*x^4])/(128*(1 + 2*x^2)) + (3917*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(640*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (1783*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(128*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) + (961*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(12 8*Sqrt[3]*Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) + (69*Sqrt[ 3]*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(16*Sqrt[(1 + x^2 )/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e *x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 10.69 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x \left (192 x^{6}-1888 x^{4}+4458 x^{2}+3035\right )}{640 \left (2 x^{2}+1\right )^{2}}-\frac {719 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{1024 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3917 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{1280 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2617 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{512 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(192\) |
| default | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{64 \left (2 x^{2}+1\right )^{2}}+\frac {649 x \sqrt {3 x^{4}+5 x^{2}+2}}{128 \left (2 x^{2}+1\right )}-\frac {19263 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5120 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3917 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{1280 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2617 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{512 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{40}-\frac {13 x \sqrt {3 x^{4}+5 x^{2}+2}}{16}\) | \(224\) |
| elliptic | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{64 \left (2 x^{2}+1\right )^{2}}+\frac {649 x \sqrt {3 x^{4}+5 x^{2}+2}}{128 \left (2 x^{2}+1\right )}-\frac {19263 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5120 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3917 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{1280 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2617 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{512 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{40}-\frac {13 x \sqrt {3 x^{4}+5 x^{2}+2}}{16}\) | \(224\) |
Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^3,x,method=_RETURNVERBOS E)
Output:
1/640*(3*x^4+5*x^2+2)^(1/2)*x*(192*x^6-1888*x^4+4458*x^2+3035)/(2*x^2+1)^2 -719/1024*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticF( I*x,1/2*6^(1/2))-3917/1280*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2) ^(1/2)*(EllipticF(I*x,1/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))-2617/512*I* (x^2+1)^(1/2)*(1+3/2*x^2)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticPi(I*x,2,1/2 *I*(-3)^(1/2)*2^(1/2))
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^3,x, algorithm="fr icas")
Output:
integral((3*x^8 - 16*x^6 - 21*x^4 + 6*x^2 + 8)*sqrt(3*x^4 + 5*x^2 + 2)/(8* x^6 + 12*x^4 + 6*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}} \left (x^{4} - 7 x^{2} + 4\right )}{\left (2 x^{2} + 1\right )^{3}}\, dx \] Input:
integrate((x**4-7*x**2+4)*(3*x**4+5*x**2+2)**(3/2)/(2*x**2+1)**3,x)
Output:
Integral(((x**2 + 1)*(3*x**2 + 2))**(3/2)*(x**4 - 7*x**2 + 4)/(2*x**2 + 1) **3, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^3,x, algorithm="ma xima")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^3, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^3,x, algorithm="gi ac")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^3, x)
Timed out. \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\int \frac {\left (x^4-7\,x^2+4\right )\,{\left (3\,x^4+5\,x^2+2\right )}^{3/2}}{{\left (2\,x^2+1\right )}^3} \,d x \] Input:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1)^3,x)
Output:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1)^3, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^3} \, dx=\frac {396 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}-3894 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}-33697 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}-68576 \sqrt {3 x^{4}+5 x^{2}+2}\, x +205344 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{4}+205344 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{2}+51336 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right )+427744 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}+427744 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+106936 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-65724 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{4}-65724 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{2}-16431 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right )+320808 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}+320808 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+80202 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-320808 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}-320808 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}-80202 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )}{5280 x^{4}+5280 x^{2}+1320} \] Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^3,x)
Output:
(396*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 3894*sqrt(3*x**4 + 5*x**2 + 2)*x**5 - 33697*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 68576*sqrt(3*x**4 + 5*x**2 + 2)*x + 205344*int(sqrt(3*x**4 + 5*x**2 + 2)/(24*x**10 + 76*x**8 + 94*x**6 + 57 *x**4 + 17*x**2 + 2),x)*x**4 + 205344*int(sqrt(3*x**4 + 5*x**2 + 2)/(24*x* *10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**2 + 51336*int(sqrt( 3*x**4 + 5*x**2 + 2)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2 ),x) + 427744*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**4 + 427744*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32 *x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 + 106936*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x) - 65724*int((sqrt(3*x** 4 + 5*x**2 + 2)*x**6)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**4 - 65724*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(24*x**10 + 76*x** 8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**2 - 16431*int((sqrt(3*x**4 + 5* x**2 + 2)*x**6)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x) + 320808*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**4 + 320808*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12* x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 + 80202*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x) - 320808 *int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x* *2 + 2),x)*x**4 - 320808*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 ...