Integrand size = 25, antiderivative size = 326 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {133 x \left (5+\sqrt {13}+2 x^2\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {266 \sqrt {3+5 x^2+x^4}}{1053 x}+\frac {133 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{351 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
1/39*(-8*x^2-7)/x^3/(x^4+5*x^2+3)^(1/2)-133/1053*x*(5+2*x^2+13^(1/2))/(x^4 +5*x^2+3)^(1/2)-5/351*(x^4+5*x^2+3)^(1/2)/x^3+266/1053*(x^4+5*x^2+3)^(1/2) /x+133/6318*(1/(36+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1 /2)*EllipticE(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2),1/6*( -78+30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*(30+6*13^(1/2))^(1/2)*((6+x^2 *(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)-5/351*(1/(3 6+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1/2)*EllipticF(x*( 30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2),1/6*(-78+30*13^(1/2))^ (1/2))*(6+x^2*(5+13^(1/2)))*((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1 /2)/(x^4+5*x^2+3)^(1/2)/(30+6*13^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 10.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-468+1014 x^2+2630 x^4+532 x^6-133 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-650+133 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{2106 x^3 \sqrt {3+5 x^2+x^4}} \]
(-468 + 1014*x^2 + 2630*x^4 + 532*x^6 - (133*I)*Sqrt[2]*(-5 + Sqrt[13])*x^ 3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2] *EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I *Sqrt[2]*(-650 + 133*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt [13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13] )]*x], 19/6 + (5*Sqrt[13])/6])/(2106*x^3*Sqrt[3 + 5*x^2 + x^4])
Time = 0.45 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1600, 25, 1604, 1604, 25, 1503, 1412, 1455}
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.
| \(\displaystyle \int \frac {3 x^2+2}{x^4 \left (x^4+5 x^2+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1600 |
| \(\displaystyle -\frac {1}{39} \int -\frac {5-24 x^2}{x^4 \sqrt {x^4+5 x^2+3}}dx-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{39} \int \frac {5-24 x^2}{x^4 \sqrt {x^4+5 x^2+3}}dx-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {1}{39} \left (-\frac {1}{9} \int \frac {5 x^2+266}{x^2 \sqrt {x^4+5 x^2+3}}dx-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {1}{39} \left (\frac {1}{9} \left (\frac {1}{3} \int -\frac {266 x^2+15}{\sqrt {x^4+5 x^2+3}}dx+\frac {266 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{39} \left (\frac {1}{9} \left (\frac {266 \sqrt {x^4+5 x^2+3}}{3 x}-\frac {1}{3} \int \frac {266 x^2+15}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{39} \left (\frac {1}{9} \left (\frac {1}{3} \left (-15 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx-266 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )+\frac {266 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{39} \left (\frac {1}{9} \left (\frac {1}{3} \left (-266 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx-\frac {5 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )+\frac {266 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{39} \left (\frac {1}{9} \left (\frac {1}{3} \left (-\frac {5 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-266 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )+\frac {266 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {5 \sqrt {x^4+5 x^2+3}}{9 x^3}\right )-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}}\) |
-1/39*(7 + 8*x^2)/(x^3*Sqrt[3 + 5*x^2 + x^4]) + ((-5*Sqrt[3 + 5*x^2 + x^4] )/(9*x^3) + ((266*Sqrt[3 + 5*x^2 + x^4])/(3*x) + (-266*((x*(5 + Sqrt[13] + 2*x^2))/(2*Sqrt[3 + 5*x^2 + x^4]) - (Sqrt[(5 + Sqrt[13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Ellipt icE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2*Sqrt[3 + 5 *x^2 + x^4])) - (5*Sqrt[3/(2*(5 + Sqrt[13]))]*Sqrt[(6 + (5 - Sqrt[13])*x^2 )/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[ArcTan[Sqrt [(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4])/3)/9) /39
3.3.3.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q )*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[ (b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^2 + c*x^4)^(p + 1) *((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2)/(2*a*f*(p + 1)*(b^2 - 4*a *c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^m*(a + b*x^2 + c *x^4)^(p + 1)*Simp[d*(b^2*(m + 2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Int egerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Time = 2.98 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {266 x^{6}+1315 x^{4}+507 x^{2}-234}{1053 x^{3} \sqrt {x^{4}+5 x^{2}+3}}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(228\) |
| elliptic | \(-\frac {2 \left (\frac {11}{702} x^{3}+\frac {17}{351} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{27 x^{3}}+\frac {23 \sqrt {x^{4}+5 x^{2}+3}}{81 x}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(251\) |
| default | \(-\frac {6 \left (\frac {19}{234} x^{3}+\frac {40}{117} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}+\frac {23 \sqrt {x^{4}+5 x^{2}+3}}{81 x}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (-\frac {40}{351} x^{3}-\frac {343}{702} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{27 x^{3}}\) | \(274\) |
1/1053*(266*x^6+1315*x^4+507*x^2-234)/x^3/(x^4+5*x^2+3)^(1/2)-10/117/(-30+ 6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2)) *x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)*EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6 *3^(1/2)+1/6*39^(1/2))+1064/117/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/ 2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)/(5+13 ^(1/2))*(EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))- EllipticE(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))
Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.63 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {266 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )} - 5 \, \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (251 \, \sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )} - 1405 \, \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, {\left (266 \, x^{6} + 1315 \, x^{4} + 507 \, x^{2} - 234\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{37908 \, {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}} \]
1/37908*(266*(sqrt(13)*sqrt(6)*sqrt(3)*(x^7 + 5*x^5 + 3*x^3) - 5*sqrt(6)*s qrt(3)*(x^7 + 5*x^5 + 3*x^3))*sqrt(sqrt(13) - 5)*elliptic_e(arcsin(1/6*sqr t(6)*x*sqrt(sqrt(13) - 5)), 5/6*sqrt(13) + 19/6) - (251*sqrt(13)*sqrt(6)*s qrt(3)*(x^7 + 5*x^5 + 3*x^3) - 1405*sqrt(6)*sqrt(3)*(x^7 + 5*x^5 + 3*x^3)) *sqrt(sqrt(13) - 5)*elliptic_f(arcsin(1/6*sqrt(6)*x*sqrt(sqrt(13) - 5)), 5 /6*sqrt(13) + 19/6) + 36*(266*x^6 + 1315*x^4 + 507*x^2 - 234)*sqrt(x^4 + 5 *x^2 + 3))/(x^7 + 5*x^5 + 3*x^3)
\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3 x^{2} + 2}{x^{4} \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3\,x^2+2}{x^4\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]