\(\int \frac {1}{(2+2 x^2-3 x^4)^{3/2}} \, dx\) [176]

Optimal result

Integrand size = 16, antiderivative size = 127 \[ \int \frac {1}{\left (2+2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {x \left (8-3 x^2\right )}{28 \sqrt {2+2 x^2-3 x^4}}+\frac {1}{28} \sqrt {-1+\sqrt {7}} E\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )+\frac {1}{4} \sqrt {\frac {1}{7} \left (-1+\sqrt {7}\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right ) \] Output:

1/28*x*(-3*x^2+8)/(-3*x^4+2*x^2+2)^(1/2)+1/28*(-1+7^(1/2))^(1/2)*EllipticE 
(3^(1/2)/(1+7^(1/2))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))+1/28*(-7+7*7^(1 
/2))^(1/2)*EllipticF(3^(1/2)/(1+7^(1/2))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1 
/2))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (2+2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {1}{28} \left (\frac {x \left (8-3 x^2\right )}{\sqrt {2+2 x^2-3 x^4}}+i \sqrt {1+\sqrt {7}} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )-\frac {i \left (7+\sqrt {7}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}}\right ) \] Input:

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

Output:

((x*(8 - 3*x^2))/Sqrt[2 + 2*x^2 - 3*x^4] + I*Sqrt[1 + Sqrt[7]]*EllipticE[I 
*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3] - (I*(7 + Sqrt[7])*E 
llipticF[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3])/Sqrt[1 + 
Sqrt[7]])/28
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}-\frac {1}{56} \int -\frac {6 \left (x^2+2\right )}{\sqrt {-3 x^4+2 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{28} \int \frac {x^2+2}{\sqrt {-3 x^4+2 x^2+2}}dx+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {3}{14} \sqrt {3} \int \frac {x^2+2}{2 \sqrt {-3 x^2+\sqrt {7}+1} \sqrt {3 x^2+\sqrt {7}-1}}dx+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{28} \sqrt {3} \int \frac {x^2+2}{\sqrt {-3 x^2+\sqrt {7}+1} \sqrt {3 x^2+\sqrt {7}-1}}dx+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {1}{3} \left (7-\sqrt {7}\right ) \int \frac {1}{\sqrt {-3 x^2+\sqrt {7}+1} \sqrt {3 x^2+\sqrt {7}-1}}dx+\frac {1}{3} \int \frac {\sqrt {3 x^2+\sqrt {7}-1}}{\sqrt {-3 x^2+\sqrt {7}+1}}dx\right )+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {1}{3} \int \frac {\sqrt {3 x^2+\sqrt {7}-1}}{\sqrt {-3 x^2+\sqrt {7}+1}}dx+\frac {\left (7-\sqrt {7}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{3 \sqrt {3 \left (\sqrt {7}-1\right )}}\right )+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {\left (7-\sqrt {7}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{3 \sqrt {3 \left (\sqrt {7}-1\right )}}+\frac {1}{3} \sqrt {\frac {1}{3} \left (\sqrt {7}-1\right )} E\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )\right )+\frac {x \left (8-3 x^2\right )}{28 \sqrt {-3 x^4+2 x^2+2}}\)

Input:

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

Output:

(x*(8 - 3*x^2))/(28*Sqrt[2 + 2*x^2 - 3*x^4]) + (3*Sqrt[3]*((Sqrt[(-1 + Sqr 
t[7])/3]*EllipticE[ArcSin[Sqrt[3/(1 + Sqrt[7])]*x], (-4 - Sqrt[7])/3])/3 + 
 ((7 - Sqrt[7])*EllipticF[ArcSin[Sqrt[3/(1 + Sqrt[7])]*x], (-4 - Sqrt[7])/ 
3])/(3*Sqrt[3*(-1 + Sqrt[7])])))/28
 

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. 229 vs. \(2 (97 ) = 194\).

Time = 1.87 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.81

method	result	size
risch	\(-\frac {x \left (3 x^{2}-8\right )}{28 \sqrt {-3 x^{4}+2 x^{2}+2}}+\frac {3 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}\, \left (2+2 \sqrt {7}\right )}\)	\(230\)
default	\(\frac {\frac {2}{7} x -\frac {3}{28} x^{3}}{\sqrt {-3 x^{4}+2 x^{2}+2}}+\frac {3 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}\, \left (2+2 \sqrt {7}\right )}\)	\(231\)
elliptic	\(\frac {\frac {2}{7} x -\frac {3}{28} x^{3}}{\sqrt {-3 x^{4}+2 x^{2}+2}}+\frac {3 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )\right )}{7 \sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}\, \left (2+2 \sqrt {7}\right )}\)	\(231\)

Input:

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

Output:

-1/28*x*(3*x^2-8)/(-3*x^4+2*x^2+2)^(1/2)+3/7/(-2+2*7^(1/2))^(1/2)*(1-(-1/2 
+1/2*7^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*7^(1/2))*x^2)^(1/2)/(-3*x^4+2*x^2+2) 
^(1/2)*EllipticF(1/2*(-2+2*7^(1/2))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))- 
6/7/(-2+2*7^(1/2))^(1/2)*(1-(-1/2+1/2*7^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*7^( 
1/2))*x^2)^(1/2)/(-3*x^4+2*x^2+2)^(1/2)/(2+2*7^(1/2))*(EllipticF(1/2*(-2+2 
*7^(1/2))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))-EllipticE(1/2*(-2+2*7^(1/2 
))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (2+2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {{\left (\sqrt {7} \sqrt {2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )} - \sqrt {2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + {\left (\sqrt {7} \sqrt {2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )} + 3 \, \sqrt {2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + 2 \, \sqrt {-3 \, x^{4} + 2 \, x^{2} + 2} {\left (3 \, x^{3} - 8 \, x\right )}}{56 \, {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}} \] Input:

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

Output:

1/56*((sqrt(7)*sqrt(2)*(3*x^4 - 2*x^2 - 2) - sqrt(2)*(3*x^4 - 2*x^2 - 2))* 
sqrt(1/2*sqrt(7) - 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(7) - 1/2)), -1/3 
*sqrt(7) - 4/3) + (sqrt(7)*sqrt(2)*(3*x^4 - 2*x^2 - 2) + 3*sqrt(2)*(3*x^4 
- 2*x^2 - 2))*sqrt(1/2*sqrt(7) - 1/2)*elliptic_f(arcsin(x*sqrt(1/2*sqrt(7) 
 - 1/2)), -1/3*sqrt(7) - 4/3) + 2*sqrt(-3*x^4 + 2*x^2 + 2)*(3*x^3 - 8*x))/ 
(3*x^4 - 2*x^2 - 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 x^{4}+2 x^{2}+2}}{9 x^{8}-12 x^{6}-8 x^{4}+8 x^{2}+4}d x \] Input:

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

Output:

int(sqrt( - 3*x**4 + 2*x**2 + 2)/(9*x**8 - 12*x**6 - 8*x**4 + 8*x**2 + 4), 
x)