Integrand size = 25, antiderivative size = 331 \[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=-\frac {49949 x \left (5+\sqrt {13}+2 x^2\right )}{3465 \sqrt {3+5 x^2+x^4}}+\frac {353}{99} x \sqrt {3+5 x^2+x^4}-\frac {x^3 \left (911+890 x^2\right ) \sqrt {3+5 x^2+x^4}}{1155}+\frac {1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {49949 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3465 \sqrt {3+5 x^2+x^4}}-\frac {353 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{33 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
1/99*x^3*(27*x^2+67)*(x^4+5*x^2+3)^(3/2)-49949/3465*x*(5+2*x^2+13^(1/2))/( x^4+5*x^2+3)^(1/2)+353/99*x*(x^4+5*x^2+3)^(1/2)-1/1155*x^3*(890*x^2+911)*( x^4+5*x^2+3)^(1/2)+49949/20790*(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)-353/33*(1/(36+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 = 7.62 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.74 \[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {2 x \left (37065+74681 x^2+69535 x^4+84962 x^6+50075 x^8+11795 x^{10}+945 x^{12}\right )-49949 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-212680+49949 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{6930 \sqrt {3+5 x^2+x^4}} \]
(2*x*(37065 + 74681*x^2 + 69535*x^4 + 84962*x^6 + 50075*x^8 + 11795*x^10 + 945*x^12) - (49949*I)*Sqrt[2]*(-5 + Sqrt[13])*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]*(-212680 + 49949*Sqr t[13])*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 ])/(6930*Sqrt[3 + 5*x^2 + x^4])
Time = 0.45 (sec) , antiderivative size = 351, 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 = {1596, 25, 1596, 25, 1602, 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.
| \(\displaystyle \int x^2 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1596 |
| \(\displaystyle \frac {1}{33} \int -x^2 \left (178 x^2+3\right ) \sqrt {x^4+5 x^2+3}dx+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{99} x^3 \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {1}{33} \int x^2 \left (178 x^2+3\right ) \sqrt {x^4+5 x^2+3}dx\) |
| \(\Big \downarrow \) 1596 |
| \(\displaystyle \frac {1}{33} \left (-\frac {1}{35} \int -\frac {x^2 \left (12355 x^2+7884\right )}{\sqrt {x^4+5 x^2+3}}dx-\frac {1}{35} \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3} x^3\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \int \frac {x^2 \left (12355 x^2+7884\right )}{\sqrt {x^4+5 x^2+3}}dx-\frac {1}{35} x^3 \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3}\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \left (\frac {12355}{3} x \sqrt {x^4+5 x^2+3}-\frac {1}{3} \int \frac {99898 x^2+37065}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {1}{35} x^3 \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3}\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \left (\frac {1}{3} \left (-37065 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx-99898 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )+\frac {12355}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1}{35} x^3 \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3}\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \left (\frac {1}{3} \left (-99898 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx-\frac {12355 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )+\frac {12355}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1}{35} x^3 \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3}\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \left (\frac {1}{3} \left (-\frac {12355 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-99898 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )+\frac {12355}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1}{35} x^3 \left (890 x^2+911\right ) \sqrt {x^4+5 x^2+3}\right )+\frac {1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3\) |
(x^3*(67 + 27*x^2)*(3 + 5*x^2 + x^4)^(3/2))/99 + (-1/35*(x^3*(911 + 890*x^ 2)*Sqrt[3 + 5*x^2 + x^4]) + ((12355*x*Sqrt[3 + 5*x^2 + x^4])/3 + (-99898*( (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 + Sqr t[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13]) /6])/(2*Sqrt[3 + 5*x^2 + x^4])) - (12355*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)/35)/33
3.2.64.3.1 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*((b*e*2 *p + c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/(c*f*(4*p + m + 1)*(m + 4*p + 3))), x] + Simp[2*(p/(c*(4*p + m + 1)*(m + 4*p + 3))) Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p - 1)*Simp[2*a*c*d*(m + 4*p + 3) - a*b*e*(m + 1) + (2*a*c *e*(4*p + m + 1) + b*c*d*(m + 4*p + 3) - b^2*e*(m + 2*p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] & & NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[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[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Time = 2.78 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {x \left (945 x^{8}+7070 x^{6}+11890 x^{4}+4302 x^{2}+12355\right ) \sqrt {x^{4}+5 x^{2}+3}}{3465}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(231\) |
| default | \(\frac {3 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{11}+\frac {202 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{99}+\frac {2378 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{693}+\frac {478 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{385}+\frac {353 x \sqrt {x^{4}+5 x^{2}+3}}{99}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(277\) |
| elliptic | \(\frac {3 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{11}+\frac {202 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{99}+\frac {2378 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{693}+\frac {478 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{385}+\frac {353 x \sqrt {x^{4}+5 x^{2}+3}}{99}-\frac {706 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{11 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {399592 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{385 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(277\) |
1/3465*x*(945*x^8+7070*x^6+11890*x^4+4302*x^2+12355)*(x^4+5*x^2+3)^(1/2)-7 06/11/(-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))+399592/385/(-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 = 144, normalized size of antiderivative = 0.44 \[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=-\frac {99898 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (87543 \, \sqrt {13} \sqrt {2} x - 561265 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 4 \, {\left (945 \, x^{10} + 7070 \, x^{8} + 11890 \, x^{6} + 4302 \, x^{4} + 12355 \, x^{2} - 99898\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{13860 \, x} \]
-1/13860*(99898*(sqrt(13)*sqrt(2)*x - 5*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elli ptic_e(arcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - (8 7543*sqrt(13)*sqrt(2)*x - 561265*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elliptic_f( arcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - 4*(945*x^ 10 + 7070*x^8 + 11890*x^6 + 4302*x^4 + 12355*x^2 - 99898)*sqrt(x^4 + 5*x^2 + 3))/x
\[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^{2} \cdot \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
\[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{2} \,d x } \]
\[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^2\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]