\(\int \frac {1}{(2+5 x^2-6 x^4)^{3/2}} \, dx\) [307]

Optimal result

Integrand size = 16, antiderivative size = 133 \[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\frac {x \left (49-30 x^2\right )}{146 \sqrt {2+5 x^2-6 x^4}}+\frac {5}{146} \sqrt {\frac {1}{2} \left (-5+\sqrt {73}\right )} E\left (\arcsin \left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right )+\frac {1}{2} \sqrt {\frac {1}{146} \left (-5+\sqrt {73}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right ),\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right ) \] Output:

1/146*x*(-30*x^2+49)/(-6*x^4+5*x^2+2)^(1/2)+5/292*(-10+2*73^(1/2))^(1/2)*E 
llipticE(2*3^(1/2)/(5+73^(1/2))^(1/2)*x,5/12*I*3^(1/2)+1/12*I*219^(1/2))+1 
/292*(-730+146*73^(1/2))^(1/2)*EllipticF(2*3^(1/2)/(5+73^(1/2))^(1/2)*x,5/ 
12*I*3^(1/2)+1/12*I*219^(1/2))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.66 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\frac {1}{292} \left (\frac {98 x}{\sqrt {2+5 x^2-6 x^4}}-\frac {60 x^3}{\sqrt {2+5 x^2-6 x^4}}+5 i \sqrt {2 \left (5+\sqrt {73}\right )} E\left (i \text {arcsinh}\left (2 \sqrt {\frac {3}{-5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49+5 \sqrt {73}\right )\right )-i \sqrt {\frac {2}{5+\sqrt {73}}} \left (73+5 \sqrt {73}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (2 \sqrt {\frac {3}{-5+\sqrt {73}}} x\right ),\frac {1}{24} \left (-49+5 \sqrt {73}\right )\right )\right ) \] Input:

Integrate[(2 + 5*x^2 - 6*x^4)^(-3/2),x]
 

Output:

((98*x)/Sqrt[2 + 5*x^2 - 6*x^4] - (60*x^3)/Sqrt[2 + 5*x^2 - 6*x^4] + (5*I) 
*Sqrt[2*(5 + Sqrt[73])]*EllipticE[I*ArcSinh[2*Sqrt[3/(-5 + Sqrt[73])]*x], 
(-49 + 5*Sqrt[73])/24] - I*Sqrt[2/(5 + Sqrt[73])]*(73 + 5*Sqrt[73])*Ellipt 
icF[I*ArcSinh[2*Sqrt[3/(-5 + Sqrt[73])]*x], (-49 + 5*Sqrt[73])/24])/292
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1494, 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.

\(\displaystyle \int \frac {1}{\left (-6 x^4+5 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}-\frac {1}{146} \int -\frac {6 \left (5 x^2+4\right )}{\sqrt {-6 x^4+5 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{73} \int \frac {5 x^2+4}{\sqrt {-6 x^4+5 x^2+2}}dx+\frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {6}{73} \sqrt {6} \int \frac {5 x^2+4}{\sqrt {-12 x^2+\sqrt {73}+5} \sqrt {12 x^2+\sqrt {73}-5}}dx+\frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {6}{73} \sqrt {6} \left (\frac {1}{12} \left (73-5 \sqrt {73}\right ) \int \frac {1}{\sqrt {-12 x^2+\sqrt {73}+5} \sqrt {12 x^2+\sqrt {73}-5}}dx+\frac {5}{12} \int \frac {\sqrt {12 x^2+\sqrt {73}-5}}{\sqrt {-12 x^2+\sqrt {73}+5}}dx\right )+\frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {6}{73} \sqrt {6} \left (\frac {5}{12} \int \frac {\sqrt {12 x^2+\sqrt {73}-5}}{\sqrt {-12 x^2+\sqrt {73}+5}}dx+\frac {\left (73-5 \sqrt {73}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right ),\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right )}{24 \sqrt {3 \left (\sqrt {73}-5\right )}}\right )+\frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6}{73} \sqrt {6} \left (\frac {\left (73-5 \sqrt {73}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right ),\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right )}{24 \sqrt {3 \left (\sqrt {73}-5\right )}}+\frac {5}{24} \sqrt {\frac {1}{3} \left (\sqrt {73}-5\right )} E\left (\arcsin \left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right )\right )+\frac {x \left (49-30 x^2\right )}{146 \sqrt {-6 x^4+5 x^2+2}}\)

Input:

Int[(2 + 5*x^2 - 6*x^4)^(-3/2),x]
 

Output:

(x*(49 - 30*x^2))/(146*Sqrt[2 + 5*x^2 - 6*x^4]) + (6*Sqrt[6]*((5*Sqrt[(-5 
+ Sqrt[73])/3]*EllipticE[ArcSin[2*Sqrt[3/(5 + Sqrt[73])]*x], (-49 - 5*Sqrt 
[73])/24])/24 + ((73 - 5*Sqrt[73])*EllipticF[ArcSin[2*Sqrt[3/(5 + Sqrt[73] 
)]*x], (-49 - 5*Sqrt[73])/24])/(24*Sqrt[3*(-5 + Sqrt[73])])))/73
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 399

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])))))
 

rule 1405

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]
 

rule 1494

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (101 ) = 202\).

Time = 2.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.64

method	result	size
risch	\(-\frac {x \left (30 x^{2}-49\right )}{146 \sqrt {-6 x^{4}+5 x^{2}+2}}+\frac {24 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}}-\frac {120 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}\, \left (5+\sqrt {73}\right )}\)	\(218\)
default	\(\frac {\frac {49}{146} x -\frac {15}{73} x^{3}}{\sqrt {-6 x^{4}+5 x^{2}+2}}+\frac {24 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}}-\frac {120 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}\, \left (5+\sqrt {73}\right )}\)	\(219\)
elliptic	\(\frac {\frac {49}{146} x -\frac {15}{73} x^{3}}{\sqrt {-6 x^{4}+5 x^{2}+2}}+\frac {24 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}}-\frac {120 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-5+\sqrt {73}}}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )\right )}{73 \sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}\, \left (5+\sqrt {73}\right )}\)	\(219\)

Input:

int(1/(-6*x^4+5*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/146*x*(30*x^2-49)/(-6*x^4+5*x^2+2)^(1/2)+24/73/(-5+73^(1/2))^(1/2)*(1-( 
-5/4+1/4*73^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*73^(1/2))*x^2)^(1/2)/(-6*x^4+5* 
x^2+2)^(1/2)*EllipticF(1/2*x*(-5+73^(1/2))^(1/2),5/12*I*3^(1/2)+1/12*I*219 
^(1/2))-120/73/(-5+73^(1/2))^(1/2)*(1-(-5/4+1/4*73^(1/2))*x^2)^(1/2)*(1-(- 
5/4-1/4*73^(1/2))*x^2)^(1/2)/(-6*x^4+5*x^2+2)^(1/2)/(5+73^(1/2))*(Elliptic 
F(1/2*x*(-5+73^(1/2))^(1/2),5/12*I*3^(1/2)+1/12*I*219^(1/2))-EllipticE(1/2 
*x*(-5+73^(1/2))^(1/2),5/12*I*3^(1/2)+1/12*I*219^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\frac {5 \, {\left (\sqrt {73} \sqrt {2} {\left (6 \, x^{4} - 5 \, x^{2} - 2\right )} - 5 \, \sqrt {2} {\left (6 \, x^{4} - 5 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} - 5} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} - 5}\right )\,|\,-\frac {5}{24} \, \sqrt {73} - \frac {49}{24}) - {\left (\sqrt {73} \sqrt {2} {\left (6 \, x^{4} - 5 \, x^{2} - 2\right )} - 45 \, \sqrt {2} {\left (6 \, x^{4} - 5 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} - 5} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} - 5}\right )\,|\,-\frac {5}{24} \, \sqrt {73} - \frac {49}{24}) + 8 \, \sqrt {-6 \, x^{4} + 5 \, x^{2} + 2} {\left (30 \, x^{3} - 49 \, x\right )}}{1168 \, {\left (6 \, x^{4} - 5 \, x^{2} - 2\right )}} \] Input:

integrate(1/(-6*x^4+5*x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

1/1168*(5*(sqrt(73)*sqrt(2)*(6*x^4 - 5*x^2 - 2) - 5*sqrt(2)*(6*x^4 - 5*x^2 
 - 2))*sqrt(sqrt(73) - 5)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(73) - 5)), -5/ 
24*sqrt(73) - 49/24) - (sqrt(73)*sqrt(2)*(6*x^4 - 5*x^2 - 2) - 45*sqrt(2)* 
(6*x^4 - 5*x^2 - 2))*sqrt(sqrt(73) - 5)*elliptic_f(arcsin(1/2*x*sqrt(sqrt( 
73) - 5)), -5/24*sqrt(73) - 49/24) + 8*sqrt(-6*x^4 + 5*x^2 + 2)*(30*x^3 - 
49*x))/(6*x^4 - 5*x^2 - 2)
 

Sympy [F]

\[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 6 x^{4} + 5 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-6*x**4+5*x**2+2)**(3/2),x)
 

Output:

Integral((-6*x**4 + 5*x**2 + 2)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-6 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-6*x^4+5*x^2+2)^(3/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((-6*x^4 + 5*x^2 + 2)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-6 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-6*x^4+5*x^2+2)^(3/2),x, algorithm="giac")
 

Output:

integrate((-6*x^4 + 5*x^2 + 2)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-6\,x^4+5\,x^2+2\right )}^{3/2}} \,d x \] Input:

int(1/(5*x^2 - 6*x^4 + 2)^(3/2),x)
 

Output:

int(1/(5*x^2 - 6*x^4 + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (2+5 x^2-6 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-6 x^{4}+5 x^{2}+2}}{36 x^{8}-60 x^{6}+x^{4}+20 x^{2}+4}d x \] Input:

int(1/(-6*x^4+5*x^2+2)^(3/2),x)
 

Output:

int(sqrt( - 6*x**4 + 5*x**2 + 2)/(36*x**8 - 60*x**6 + x**4 + 20*x**2 + 4), 
x)