Integrand size = 16, antiderivative size = 236 \[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 x}{\sqrt {57} \left (9-\sqrt {57}\right ) \sqrt {3+9 x^2+2 x^4}}+\frac {2 \sqrt {\frac {6}{9+\sqrt {57}}} \sqrt {3+9 x^2+2 x^4} E\left (\arctan \left (\frac {2 x}{\sqrt {9+\sqrt {57}}}\right )|\frac {1}{4} \left (-19-3 \sqrt {57}\right )\right )}{19 \sqrt {\frac {6}{9+\sqrt {57}}+x^2} \sqrt {9+\sqrt {57}+4 x^2}}-\frac {\sqrt {9+\sqrt {57}} \sqrt {6+\left (9-\sqrt {57}\right ) x^2} \sqrt {6+\left (9+\sqrt {57}\right ) x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (9-\sqrt {57}\right )} x\right ),\frac {1}{4} \left (-19-3 \sqrt {57}\right )\right )}{171 \sqrt {3+9 x^2+2 x^4}} \] Output:
4/57*x*57^(1/2)/(9-57^(1/2))/(2*x^4+9*x^2+3)^(1/2)+2/19*6^(1/2)/(9+57^(1/2 ))^(1/2)*(2*x^4+9*x^2+3)^(1/2)*EllipticE(2*x/(9+57^(1/2))^(1/2)/(1+4*x^2/( 9+57^(1/2)))^(1/2),1/2*(-19-3*57^(1/2))^(1/2))/(6/(9+57^(1/2))+x^2)^(1/2)/ (9+57^(1/2)+4*x^2)^(1/2)-1/171*(9+57^(1/2))^(1/2)*(6+(9-57^(1/2))*x^2)^(1/ 2)*(6+(9+57^(1/2))*x^2)^(1/2)*InverseJacobiAM(arctan(1/6*(54-6*57^(1/2))^( 1/2)*x),1/2*(-19-3*57^(1/2))^(1/2))/(2*x^4+9*x^2+3)^(1/2)
Result contains complex when optimal does not.
Time = 6.18 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {9-\sqrt {57}} x \left (23+6 x^2\right )-3 i \left (-9+\sqrt {57}\right ) \sqrt {9-\sqrt {57}+4 x^2} \sqrt {9+\sqrt {57}+4 x^2} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {9+\sqrt {57}}}\right )|\frac {23}{4}+\frac {3 \sqrt {57}}{4}\right )+i \left (-19+3 \sqrt {57}\right ) \sqrt {9-\sqrt {57}+4 x^2} \sqrt {9+\sqrt {57}+4 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {9+\sqrt {57}}}\right ),\frac {23}{4}+\frac {3 \sqrt {57}}{4}\right )}{228 \sqrt {-\left (\left (-9+\sqrt {57}\right ) \left (3+9 x^2+2 x^4\right )\right )}} \] Input:
Integrate[(3 + 9*x^2 + 2*x^4)^(-3/2),x]
Output:
(4*Sqrt[9 - Sqrt[57]]*x*(23 + 6*x^2) - (3*I)*(-9 + Sqrt[57])*Sqrt[9 - Sqrt [57] + 4*x^2]*Sqrt[9 + Sqrt[57] + 4*x^2]*EllipticE[I*ArcSinh[(2*x)/Sqrt[9 + Sqrt[57]]], 23/4 + (3*Sqrt[57])/4] + I*(-19 + 3*Sqrt[57])*Sqrt[9 - Sqrt[ 57] + 4*x^2]*Sqrt[9 + Sqrt[57] + 4*x^2]*EllipticF[I*ArcSinh[(2*x)/Sqrt[9 + Sqrt[57]]], 23/4 + (3*Sqrt[57])/4])/(228*Sqrt[-((-9 + Sqrt[57])*(3 + 9*x^ 2 + 2*x^4))])
Time = 0.72 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 27, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^4+9 x^2+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (6 x^2+23\right )}{57 \sqrt {2 x^4+9 x^2+3}}-\frac {1}{171} \int \frac {6 \left (3 x^2+2\right )}{\sqrt {2 x^4+9 x^2+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (6 x^2+23\right )}{57 \sqrt {2 x^4+9 x^2+3}}-\frac {2}{57} \int \frac {3 x^2+2}{\sqrt {2 x^4+9 x^2+3}}dx\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {x \left (6 x^2+23\right )}{57 \sqrt {2 x^4+9 x^2+3}}-\frac {2}{57} \left (2 \int \frac {1}{\sqrt {2 x^4+9 x^2+3}}dx+3 \int \frac {x^2}{\sqrt {2 x^4+9 x^2+3}}dx\right )\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {x \left (6 x^2+23\right )}{57 \sqrt {2 x^4+9 x^2+3}}-\frac {2}{57} \left (3 \int \frac {x^2}{\sqrt {2 x^4+9 x^2+3}}dx+\frac {\sqrt {\frac {2}{3 \left (9+\sqrt {57}\right )}} \sqrt {\frac {\left (9-\sqrt {57}\right ) x^2+6}{\left (9+\sqrt {57}\right ) x^2+6}} \left (\left (9+\sqrt {57}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right ),\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{\sqrt {2 x^4+9 x^2+3}}\right )\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {x \left (6 x^2+23\right )}{57 \sqrt {2 x^4+9 x^2+3}}-\frac {2}{57} \left (\frac {\sqrt {\frac {2}{3 \left (9+\sqrt {57}\right )}} \sqrt {\frac {\left (9-\sqrt {57}\right ) x^2+6}{\left (9+\sqrt {57}\right ) x^2+6}} \left (\left (9+\sqrt {57}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right ),\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{\sqrt {2 x^4+9 x^2+3}}+3 \left (\frac {x \left (4 x^2+\sqrt {57}+9\right )}{4 \sqrt {2 x^4+9 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} \sqrt {\frac {\left (9-\sqrt {57}\right ) x^2+6}{\left (9+\sqrt {57}\right ) x^2+6}} \left (\left (9+\sqrt {57}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right )|\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{4 \sqrt {2 x^4+9 x^2+3}}\right )\right )\) |
Input:
Int[(3 + 9*x^2 + 2*x^4)^(-3/2),x]
Output:
(x*(23 + 6*x^2))/(57*Sqrt[3 + 9*x^2 + 2*x^4]) - (2*(3*((x*(9 + Sqrt[57] + 4*x^2))/(4*Sqrt[3 + 9*x^2 + 2*x^4]) - (Sqrt[(9 + Sqrt[57])/6]*Sqrt[(6 + (9 - Sqrt[57])*x^2)/(6 + (9 + Sqrt[57])*x^2)]*(6 + (9 + Sqrt[57])*x^2)*Ellip ticE[ArcTan[Sqrt[(9 + Sqrt[57])/6]*x], (-19 + 3*Sqrt[57])/4])/(4*Sqrt[3 + 9*x^2 + 2*x^4])) + (Sqrt[2/(3*(9 + Sqrt[57]))]*Sqrt[(6 + (9 - Sqrt[57])*x^ 2)/(6 + (9 + Sqrt[57])*x^2)]*(6 + (9 + Sqrt[57])*x^2)*EllipticF[ArcTan[Sqr t[(9 + Sqrt[57])/6]*x], (-19 + 3*Sqrt[57])/4])/Sqrt[3 + 9*x^2 + 2*x^4]))/5 7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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 = 2.72 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {x \left (6 x^{2}+23\right )}{57 \sqrt {2 x^{4}+9 x^{2}+3}}-\frac {8 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}}+\frac {72 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}\, \left (9+\sqrt {57}\right )}\) | \(222\) |
| default | \(-\frac {4 \left (-\frac {23}{228} x -\frac {1}{38} x^{3}\right )}{\sqrt {2 x^{4}+9 x^{2}+3}}-\frac {8 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}}+\frac {72 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}\, \left (9+\sqrt {57}\right )}\) | \(223\) |
| elliptic | \(-\frac {4 \left (-\frac {23}{228} x -\frac {1}{38} x^{3}\right )}{\sqrt {2 x^{4}+9 x^{2}+3}}-\frac {8 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}}+\frac {72 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-54+6 \sqrt {57}}}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )\right )}{19 \sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}\, \left (9+\sqrt {57}\right )}\) | \(223\) |
Input:
int(1/(2*x^4+9*x^2+3)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/57*x*(6*x^2+23)/(2*x^4+9*x^2+3)^(1/2)-8/19/(-54+6*57^(1/2))^(1/2)*(1-(-3 /2+1/6*57^(1/2))*x^2)^(1/2)*(1-(-3/2-1/6*57^(1/2))*x^2)^(1/2)/(2*x^4+9*x^2 +3)^(1/2)*EllipticF(1/6*x*(-54+6*57^(1/2))^(1/2),3/4*6^(1/2)+1/4*38^(1/2)) +72/19/(-54+6*57^(1/2))^(1/2)*(1-(-3/2+1/6*57^(1/2))*x^2)^(1/2)*(1-(-3/2-1 /6*57^(1/2))*x^2)^(1/2)/(2*x^4+9*x^2+3)^(1/2)/(9+57^(1/2))*(EllipticF(1/6* x*(-54+6*57^(1/2))^(1/2),3/4*6^(1/2)+1/4*38^(1/2))-EllipticE(1/6*x*(-54+6* 57^(1/2))^(1/2),3/4*6^(1/2)+1/4*38^(1/2)))
Time = 0.07 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {\frac {19}{3}} \sqrt {3} {\left (2 \, x^{4} + 9 \, x^{2} + 3\right )} - 3 \, \sqrt {3} {\left (2 \, x^{4} + 9 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} - \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} - \frac {3}{2}}\right )\,|\,\frac {9}{4} \, \sqrt {\frac {19}{3}} + \frac {23}{4}) - {\left (\sqrt {\frac {19}{3}} \sqrt {3} {\left (2 \, x^{4} + 9 \, x^{2} + 3\right )} - 15 \, \sqrt {3} {\left (2 \, x^{4} + 9 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} - \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} - \frac {3}{2}}\right )\,|\,\frac {9}{4} \, \sqrt {\frac {19}{3}} + \frac {23}{4}) + 2 \, \sqrt {2 \, x^{4} + 9 \, x^{2} + 3} {\left (6 \, x^{3} + 23 \, x\right )}}{114 \, {\left (2 \, x^{4} + 9 \, x^{2} + 3\right )}} \] Input:
integrate(1/(2*x^4+9*x^2+3)^(3/2),x, algorithm="fricas")
Output:
1/114*(3*(sqrt(19/3)*sqrt(3)*(2*x^4 + 9*x^2 + 3) - 3*sqrt(3)*(2*x^4 + 9*x^ 2 + 3))*sqrt(1/2*sqrt(19/3) - 3/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(19/3) - 3/2)), 9/4*sqrt(19/3) + 23/4) - (sqrt(19/3)*sqrt(3)*(2*x^4 + 9*x^2 + 3) - 15*sqrt(3)*(2*x^4 + 9*x^2 + 3))*sqrt(1/2*sqrt(19/3) - 3/2)*elliptic_f(a rcsin(x*sqrt(1/2*sqrt(19/3) - 3/2)), 9/4*sqrt(19/3) + 23/4) + 2*sqrt(2*x^4 + 9*x^2 + 3)*(6*x^3 + 23*x))/(2*x^4 + 9*x^2 + 3)
\[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{4} + 9 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(2*x**4+9*x**2+3)**(3/2),x)
Output:
Integral((2*x**4 + 9*x**2 + 3)**(-3/2), x)
\[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 9 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+9*x^2+3)^(3/2),x, algorithm="maxima")
Output:
integrate((2*x^4 + 9*x^2 + 3)^(-3/2), x)
\[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 9 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+9*x^2+3)^(3/2),x, algorithm="giac")
Output:
integrate((2*x^4 + 9*x^2 + 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (2\,x^4+9\,x^2+3\right )}^{3/2}} \,d x \] Input:
int(1/(9*x^2 + 2*x^4 + 3)^(3/2),x)
Output:
int(1/(9*x^2 + 2*x^4 + 3)^(3/2), x)
\[ \int \frac {1}{\left (3+9 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}+9 x^{2}+3}}{4 x^{8}+36 x^{6}+93 x^{4}+54 x^{2}+9}d x \] Input:
int(1/(2*x^4+9*x^2+3)^(3/2),x)
Output:
int(sqrt(2*x**4 + 9*x**2 + 3)/(4*x**8 + 36*x**6 + 93*x**4 + 54*x**2 + 9),x )