Integrand size = 16, antiderivative size = 75 \[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\frac {x \left (13+3 x^2\right )}{50 \sqrt {2-x^2-3 x^4}}-\frac {1}{50} \sqrt {3} E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )+\frac {1}{10} \sqrt {3} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right ) \] Output:
1/50*x*(3*x^2+13)/(-3*x^4-x^2+2)^(1/2)-1/50*3^(1/2)*EllipticE(1/2*x*6^(1/2 ),1/3*I*6^(1/2))+1/10*EllipticF(1/2*x*6^(1/2),1/3*I*6^(1/2))*3^(1/2)
Time = 8.95 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\frac {13 x+3 x^3-\sqrt {6-9 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )+5 \sqrt {6-9 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{50 \sqrt {2-x^2-3 x^4}} \] Input:
Integrate[(2 - x^2 - 3*x^4)^(-3/2),x]
Output:
(13*x + 3*x^3 - Sqrt[6 - 9*x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[Sqrt[3/2]*x ], -2/3] + 5*Sqrt[6 - 9*x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/(50*Sqrt[2 - x^2 - 3*x^4])
Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1405, 27, 1494, 27, 399, 321, 327}
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.
| \(\displaystyle \int \frac {1}{\left (-3 x^4-x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}-\frac {1}{50} \int -\frac {3 \left (4-x^2\right )}{\sqrt {-3 x^4-x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{50} \int \frac {4-x^2}{\sqrt {-3 x^4-x^2+2}}dx+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {3}{25} \sqrt {3} \int \frac {4-x^2}{2 \sqrt {3} \sqrt {2-3 x^2} \sqrt {x^2+1}}dx+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{50} \int \frac {4-x^2}{\sqrt {2-3 x^2} \sqrt {x^2+1}}dx+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {3}{50} \left (5 \int \frac {1}{\sqrt {2-3 x^2} \sqrt {x^2+1}}dx-\int \frac {\sqrt {x^2+1}}{\sqrt {2-3 x^2}}dx\right )+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {3}{50} \left (\frac {5 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}}-\int \frac {\sqrt {x^2+1}}{\sqrt {2-3 x^2}}dx\right )+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3}{50} \left (\frac {5 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}}-\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}}\right )+\frac {x \left (3 x^2+13\right )}{50 \sqrt {-3 x^4-x^2+2}}\) |
Input:
Int[(2 - x^2 - 3*x^4)^(-3/2),x]
Output:
(x*(13 + 3*x^2))/(50*Sqrt[2 - x^2 - 3*x^4]) + (3*(-(EllipticE[ArcSin[Sqrt[ 3/2]*x], -2/3]/Sqrt[3]) + (5*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/Sqrt[3] ))/50
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]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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[((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[2*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e }, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (61 ) = 122\).
Time = 2.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}+13\right )}{50 \sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{25 \sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{100 \sqrt {-3 x^{4}-x^{2}+2}}\) | \(138\) |
| default | \(\frac {\frac {13}{50} x +\frac {3}{50} x^{3}}{\sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{25 \sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{100 \sqrt {-3 x^{4}-x^{2}+2}}\) | \(139\) |
| elliptic | \(\frac {\frac {13}{50} x +\frac {3}{50} x^{3}}{\sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{25 \sqrt {-3 x^{4}-x^{2}+2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{100 \sqrt {-3 x^{4}-x^{2}+2}}\) | \(139\) |
Input:
int(1/(-3*x^4-x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/50*x*(3*x^2+13)/(-3*x^4-x^2+2)^(1/2)+1/25*6^(1/2)*(-6*x^2+4)^(1/2)*(x^2+ 1)^(1/2)/(-3*x^4-x^2+2)^(1/2)*EllipticF(1/2*x*6^(1/2),1/3*I*6^(1/2))+1/100 *6^(1/2)*(-6*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-3*x^4-x^2+2)^(1/2)*(EllipticF(1/ 2*x*6^(1/2),1/3*I*6^(1/2))-EllipticE(1/2*x*6^(1/2),1/3*I*6^(1/2)))
Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=-\frac {3 \, \sqrt {2} \sqrt {\frac {3}{2}} {\left (3 \, x^{4} + x^{2} - 2\right )} E(\arcsin \left (\sqrt {\frac {3}{2}} x\right )\,|\,-\frac {2}{3}) - 11 \, \sqrt {2} \sqrt {\frac {3}{2}} {\left (3 \, x^{4} + x^{2} - 2\right )} F(\arcsin \left (\sqrt {\frac {3}{2}} x\right )\,|\,-\frac {2}{3}) + 2 \, \sqrt {-3 \, x^{4} - x^{2} + 2} {\left (3 \, x^{3} + 13 \, x\right )}}{100 \, {\left (3 \, x^{4} + x^{2} - 2\right )}} \] Input:
integrate(1/(-3*x^4-x^2+2)^(3/2),x, algorithm="fricas")
Output:
-1/100*(3*sqrt(2)*sqrt(3/2)*(3*x^4 + x^2 - 2)*elliptic_e(arcsin(sqrt(3/2)* x), -2/3) - 11*sqrt(2)*sqrt(3/2)*(3*x^4 + x^2 - 2)*elliptic_f(arcsin(sqrt( 3/2)*x), -2/3) + 2*sqrt(-3*x^4 - x^2 + 2)*(3*x^3 + 13*x))/(3*x^4 + x^2 - 2 )
\[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 3 x^{4} - x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-3*x**4-x**2+2)**(3/2),x)
Output:
Integral((-3*x**4 - x**2 + 2)**(-3/2), x)
\[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-3*x^4-x^2+2)^(3/2),x, algorithm="maxima")
Output:
integrate((-3*x^4 - x^2 + 2)^(-3/2), x)
\[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-3*x^4-x^2+2)^(3/2),x, algorithm="giac")
Output:
integrate((-3*x^4 - x^2 + 2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-3\,x^4-x^2+2\right )}^{3/2}} \,d x \] Input:
int(1/(2 - 3*x^4 - x^2)^(3/2),x)
Output:
int(1/(2 - 3*x^4 - x^2)^(3/2), x)
\[ \int \frac {1}{\left (2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 x^{4}-x^{2}+2}}{9 x^{8}+6 x^{6}-11 x^{4}-4 x^{2}+4}d x \] Input:
int(1/(-3*x^4-x^2+2)^(3/2),x)
Output:
int(sqrt( - 3*x**4 - x**2 + 2)/(9*x**8 + 6*x**6 - 11*x**4 - 4*x**2 + 4),x)