Integrand size = 16, antiderivative size = 75 \[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\frac {x \left (37+10 x^2\right )}{147 \sqrt {3-5 x^2-2 x^4}}-\frac {5}{49} \sqrt {\frac {2}{3}} E\left (\arcsin \left (\sqrt {2} x\right )|-\frac {1}{6}\right )+\frac {1}{7} \sqrt {\frac {2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{6}\right ) \] Output:
1/147*x*(10*x^2+37)/(-2*x^4-5*x^2+3)^(1/2)-5/147*6^(1/2)*EllipticE(x*2^(1/ 2),1/6*I*6^(1/2))+1/21*EllipticF(x*2^(1/2),1/6*I*6^(1/2))*6^(1/2)
Time = 9.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\frac {37 x+10 x^3-5 \sqrt {6-12 x^2} \sqrt {3+x^2} E\left (\arcsin \left (\sqrt {2} x\right )|-\frac {1}{6}\right )+7 \sqrt {6-12 x^2} \sqrt {3+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{6}\right )}{147 \sqrt {3-5 x^2-2 x^4}} \] Input:
Integrate[(3 - 5*x^2 - 2*x^4)^(-3/2),x]
Output:
(37*x + 10*x^3 - 5*Sqrt[6 - 12*x^2]*Sqrt[3 + x^2]*EllipticE[ArcSin[Sqrt[2] *x], -1/6] + 7*Sqrt[6 - 12*x^2]*Sqrt[3 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/6])/(147*Sqrt[3 - 5*x^2 - 2*x^4])
Time = 0.37 (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.
| \(\displaystyle \int \frac {1}{\left (-2 x^4-5 x^2+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}-\frac {1}{147} \int -\frac {2 \left (6-5 x^2\right )}{\sqrt {-2 x^4-5 x^2+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{147} \int \frac {6-5 x^2}{\sqrt {-2 x^4-5 x^2+3}}dx+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {4}{147} \sqrt {2} \int \frac {6-5 x^2}{2 \sqrt {2} \sqrt {1-2 x^2} \sqrt {x^2+3}}dx+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{147} \int \frac {6-5 x^2}{\sqrt {1-2 x^2} \sqrt {x^2+3}}dx+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {2}{147} \left (21 \int \frac {1}{\sqrt {1-2 x^2} \sqrt {x^2+3}}dx-5 \int \frac {\sqrt {x^2+3}}{\sqrt {1-2 x^2}}dx\right )+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{147} \left (7 \sqrt {\frac {3}{2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{6}\right )-5 \int \frac {\sqrt {x^2+3}}{\sqrt {1-2 x^2}}dx\right )+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{147} \left (7 \sqrt {\frac {3}{2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{6}\right )-5 \sqrt {\frac {3}{2}} E\left (\arcsin \left (\sqrt {2} x\right )|-\frac {1}{6}\right )\right )+\frac {x \left (10 x^2+37\right )}{147 \sqrt {-2 x^4-5 x^2+3}}\) |
Input:
Int[(3 - 5*x^2 - 2*x^4)^(-3/2),x]
Output:
(x*(37 + 10*x^2))/(147*Sqrt[3 - 5*x^2 - 2*x^4]) + (2*(-5*Sqrt[3/2]*Ellipti cE[ArcSin[Sqrt[2]*x], -1/6] + 7*Sqrt[3/2]*EllipticF[ArcSin[Sqrt[2]*x], -1/ 6]))/147
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. 138 vs. \(2 (59 ) = 118\).
Time = 2.77 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.85
| method | result | size |
| risch | \(\frac {x \left (10 x^{2}+37\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}\) | \(139\) |
| default | \(\frac {\frac {37}{147} x +\frac {10}{147} x^{3}}{\sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}\) | \(140\) |
| elliptic | \(\frac {\frac {37}{147} x +\frac {10}{147} x^{3}}{\sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}+\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (x \sqrt {2}, \frac {i \sqrt {6}}{6}\right )\right )}{147 \sqrt {-2 x^{4}-5 x^{2}+3}}\) | \(140\) |
Input:
int(1/(-2*x^4-5*x^2+3)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/147*x*(10*x^2+37)/(-2*x^4-5*x^2+3)^(1/2)+2/147*2^(1/2)*(-2*x^2+1)^(1/2)* (3*x^2+9)^(1/2)/(-2*x^4-5*x^2+3)^(1/2)*EllipticF(x*2^(1/2),1/6*I*6^(1/2))+ 5/147*2^(1/2)*(-2*x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-2*x^4-5*x^2+3)^(1/2)*(Ell ipticF(x*2^(1/2),1/6*I*6^(1/2))-EllipticE(x*2^(1/2),1/6*I*6^(1/2)))
Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {10 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 5 \, x^{2} - 3\right )} E(\arcsin \left (\sqrt {2} x\right )\,|\,-\frac {1}{6}) - 12 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 5 \, x^{2} - 3\right )} F(\arcsin \left (\sqrt {2} x\right )\,|\,-\frac {1}{6}) + \sqrt {-2 \, x^{4} - 5 \, x^{2} + 3} {\left (10 \, x^{3} + 37 \, x\right )}}{147 \, {\left (2 \, x^{4} + 5 \, x^{2} - 3\right )}} \] Input:
integrate(1/(-2*x^4-5*x^2+3)^(3/2),x, algorithm="fricas")
Output:
-1/147*(10*sqrt(3)*sqrt(2)*(2*x^4 + 5*x^2 - 3)*elliptic_e(arcsin(sqrt(2)*x ), -1/6) - 12*sqrt(3)*sqrt(2)*(2*x^4 + 5*x^2 - 3)*elliptic_f(arcsin(sqrt(2 )*x), -1/6) + sqrt(-2*x^4 - 5*x^2 + 3)*(10*x^3 + 37*x))/(2*x^4 + 5*x^2 - 3 )
\[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 2 x^{4} - 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-2*x**4-5*x**2+3)**(3/2),x)
Output:
Integral((-2*x**4 - 5*x**2 + 3)**(-3/2), x)
\[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} - 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-2*x^4-5*x^2+3)^(3/2),x, algorithm="maxima")
Output:
integrate((-2*x^4 - 5*x^2 + 3)^(-3/2), x)
\[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} - 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-2*x^4-5*x^2+3)^(3/2),x, algorithm="giac")
Output:
integrate((-2*x^4 - 5*x^2 + 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-2\,x^4-5\,x^2+3\right )}^{3/2}} \,d x \] Input:
int(1/(3 - 2*x^4 - 5*x^2)^(3/2),x)
Output:
int(1/(3 - 2*x^4 - 5*x^2)^(3/2), x)
\[ \int \frac {1}{\left (3-5 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-2 x^{4}-5 x^{2}+3}}{4 x^{8}+20 x^{6}+13 x^{4}-30 x^{2}+9}d x \] Input:
int(1/(-2*x^4-5*x^2+3)^(3/2),x)
Output:
int(sqrt( - 2*x**4 - 5*x**2 + 3)/(4*x**8 + 20*x**6 + 13*x**4 - 30*x**2 + 9 ),x)