Integrand size = 22, antiderivative size = 257 \[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {3 x \left (5+\sqrt {13}+2 x^2\right )}{2 \sqrt {3+5 x^2+x^4}}-\frac {\sqrt {\frac {3}{2} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {3+5 x^2+x^4}}+\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]
3/2*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3)^(1/2)+1/3*(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)-1/4*(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 = 10.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.62 \[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {i \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \left (3 \left (-5+\sqrt {13}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+\left (11-3 \sqrt {13}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )\right )}{2 \sqrt {2} \sqrt {3+5 x^2+x^4}} \]
((I/2)*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2 *x^2]*(3*(-5 + Sqrt[13])*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19 /6 + (5*Sqrt[13])/6] + (11 - 3*Sqrt[13])*EllipticF[I*ArcSinh[Sqrt[2/(5 + S qrt[13])]*x], 19/6 + (5*Sqrt[13])/6]))/(Sqrt[2]*Sqrt[3 + 5*x^2 + x^4])
Time = 0.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {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.
| \(\displaystyle \int \frac {3 x^2+2}{\sqrt {x^4+5 x^2+3}} \, dx\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle 2 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx+3 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle 3 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx+\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+3 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\) |
3*((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[1 3])/6])/(2*Sqrt[3 + 5*x^2 + x^4])) + (Sqrt[2/(3*(5 + Sqrt[13]))]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*El lipticF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4]
3.2.91.3.1 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]
Time = 0.84 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(\frac {12 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {108 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(194\) |
| elliptic | \(\frac {12 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {108 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(194\) |
12/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*1 3^(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))-108/(-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 = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.46 \[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {9 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (7 \, \sqrt {13} \sqrt {2} x - 55 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{12 \, x} \]
1/12*(9*(sqrt(13)*sqrt(2)*x - 5*sqrt(2)*x)*sqrt(sqrt(13) - 5)*elliptic_e(a rcsin(1/2*sqrt(2)*sqrt(sqrt(13) - 5)/x), 5/6*sqrt(13) + 19/6) - (7*sqrt(13 )*sqrt(2)*x - 55*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) + 36*sqrt(x^4 + 5*x^2 + 3)) /x
\[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
\[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3}} \,d x } \]
\[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3}} \,d x } \]
Timed out. \[ \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{\sqrt {x^4+5\,x^2+3}} \,d x \]