Integrand size = 25, antiderivative size = 356 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {176723 x \left (5+\sqrt {13}+2 x^2\right )}{4290 \sqrt {3+5 x^2+x^4}}-\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {3+5 x^2+x^4}}+\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {3+5 x^2+x^4}} \]
1/143*x^5*(33*x^2+71)*(x^4+5*x^2+3)^(3/2)+176723/4290*x*(5+2*x^2+13^(1/2)) /(x^4+5*x^2+3)^(1/2)-4210/429*x*(x^4+5*x^2+3)^(1/2)+1251/715*x^3*(x^4+5*x^ 2+3)^(1/2)-1/429*x^5*(272*x^2+283)*(x^4+5*x^2+3)^(1/2)+2105/429*(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^(1/2)/(5+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)-176723/25740*(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)
Result contains complex when optimal does not.
Time = 9.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.70 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {4 x \left (-63150-93991 x^2+3055 x^4+29003 x^6+39650 x^8+24635 x^{10}+6015 x^{12}+495 x^{14}\right )+176723 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-757315+176723 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{8580 \sqrt {3+5 x^2+x^4}} \]
(4*x*(-63150 - 93991*x^2 + 3055*x^4 + 29003*x^6 + 39650*x^8 + 24635*x^10 + 6015*x^12 + 495*x^14) + (176723*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqr t[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*Arc Sinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-75731 5 + 176723*Sqrt[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])/(8580*Sqrt[3 + 5*x^2 + x^4])
Time = 0.51 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1596, 25, 1596, 27, 1602, 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.
| \(\displaystyle \int x^4 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1596 |
| \(\displaystyle \frac {3}{143} \int -x^4 \left (272 x^2+69\right ) \sqrt {x^4+5 x^2+3}dx+\frac {1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \int x^4 \left (272 x^2+69\right ) \sqrt {x^4+5 x^2+3}dx\) |
| \(\Big \downarrow \) 1596 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{63} \int -\frac {21 x^4 \left (1251 x^2+794\right )}{\sqrt {x^4+5 x^2+3}}dx+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{9} x^5 \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3}-\frac {1}{3} \int \frac {x^4 \left (1251 x^2+794\right )}{\sqrt {x^4+5 x^2+3}}dx\right )\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{3} \left (\frac {1}{5} \int \frac {x^2 \left (21050 x^2+11259\right )}{\sqrt {x^4+5 x^2+3}}dx-\frac {1251}{5} x^3 \sqrt {x^4+5 x^2+3}\right )+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {21050}{3} x \sqrt {x^4+5 x^2+3}-\frac {1}{3} \int \frac {176723 x^2+63150}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {1251}{5} x^3 \sqrt {x^4+5 x^2+3}\right )+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{3} \left (-63150 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx-176723 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )+\frac {21050}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1251}{5} x^3 \sqrt {x^4+5 x^2+3}\right )+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{3} \left (-176723 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx-\frac {10525 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )+\frac {21050}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1251}{5} x^3 \sqrt {x^4+5 x^2+3}\right )+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{143} x^5 \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {3}{143} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {10525 \sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-176723 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )+\frac {21050}{3} \sqrt {x^4+5 x^2+3} x\right )-\frac {1251}{5} x^3 \sqrt {x^4+5 x^2+3}\right )+\frac {1}{9} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5\right )\) |
(x^5*(71 + 33*x^2)*(3 + 5*x^2 + x^4)^(3/2))/143 - (3*((x^5*(283 + 272*x^2) *Sqrt[3 + 5*x^2 + x^4])/9 + ((-1251*x^3*Sqrt[3 + 5*x^2 + x^4])/5 + ((21050 *x*Sqrt[3 + 5*x^2 + x^4])/3 + (-176723*((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)*EllipticE[ArcTan[Sqrt[ (5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2*Sqrt[3 + 5*x^2 + x^4])) - (10525*Sqrt[6/(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)/5)/3))/143
3.2.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 4.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {x \left (495 x^{10}+3540 x^{8}+5450 x^{6}+1780 x^{4}+3753 x^{2}-21050\right ) \sqrt {x^{4}+5 x^{2}+3}}{2145}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(236\) |
| default | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
| elliptic | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
1/2145*x*(495*x^10+3540*x^8+5450*x^6+1780*x^4+3753*x^2-21050)*(x^4+5*x^2+3 )^(1/2)+25260/143/(-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*(-3 0+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-2120676/715/(-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 = 149, normalized size of antiderivative = 0.42 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {176723 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (155673 \, \sqrt {13} \sqrt {2} x - 988865 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (495 \, x^{12} + 3540 \, x^{10} + 5450 \, x^{8} + 1780 \, x^{6} + 3753 \, x^{4} - 21050 \, x^{2} + 176723\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{8580 \, x} \]
1/8580*(176723*(sqrt(13)*sqrt(2)*x - 5*sqrt(2)*x)*sqrt(sqrt(13) - 5)*ellip tic_e(arcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - (15 5673*sqrt(13)*sqrt(2)*x - 988865*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*(495*x^ 12 + 3540*x^10 + 5450*x^8 + 1780*x^6 + 3753*x^4 - 21050*x^2 + 176723)*sqrt (x^4 + 5*x^2 + 3))/x
\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^{4} \cdot \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
Timed out. \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^4\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]