\(\int (3-2 x^2-x^4)^{3/2} \, dx\) [337]

Optimal result

Integrand size = 16, antiderivative size = 80 \[ \int \left (3-2 x^2-x^4\right )^{3/2} \, dx=\frac {2}{35} x \left (13-3 x^2\right ) \sqrt {3-2 x^2-x^4}+\frac {1}{7} x \left (3-2 x^2-x^4\right )^{3/2}-\frac {16}{5} \sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )+\frac {176}{35} \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right ) \] Output:

2/35*x*(-3*x^2+13)*(-x^4-2*x^2+3)^(1/2)+1/7*x*(-x^4-2*x^2+3)^(3/2)-16/5*El 
lipticE(x,1/3*I*3^(1/2))*3^(1/2)+176/35*EllipticF(x,1/3*I*3^(1/2))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.65 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \left (3-2 x^2-x^4\right )^{3/2} \, dx=\frac {123 x-130 x^3-24 x^5+26 x^7+5 x^9-112 i \sqrt {3-2 x^2-x^4} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-80 i \sqrt {3-2 x^2-x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )}{35 \sqrt {3-2 x^2-x^4}} \] Input:

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

Output:

(123*x - 130*x^3 - 24*x^5 + 26*x^7 + 5*x^9 - (112*I)*Sqrt[3 - 2*x^2 - x^4] 
*EllipticE[I*ArcSinh[x/Sqrt[3]], -3] - (80*I)*Sqrt[3 - 2*x^2 - x^4]*Ellipt 
icF[I*ArcSinh[x/Sqrt[3]], -3])/(35*Sqrt[3 - 2*x^2 - x^4])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1404, 27, 1490, 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.

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

\(\Big \downarrow \) 1404

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \int \left (3-x^2\right ) \sqrt {-x^4-2 x^2+3}dx+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {6}{7} \left (\frac {1}{15} x \left (13-3 x^2\right ) \sqrt {-x^4-2 x^2+3}-\frac {1}{15} \int -\frac {8 \left (12-7 x^2\right )}{\sqrt {-x^4-2 x^2+3}}dx\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 x^2}{\sqrt {-x^4-2 x^2+3}}dx+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {6}{7} \left (\frac {16}{15} \int \frac {12-7 x^2}{2 \sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 x^2}{\sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (33 \int \frac {1}{\sqrt {1-x^2} \sqrt {x^2+3}}dx-7 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-7 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-7 \sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )\right )+\frac {1}{15} x \sqrt {-x^4-2 x^2+3} \left (13-3 x^2\right )\right )+\frac {1}{7} x \left (-x^4-2 x^2+3\right )^{3/2}\)

Input:

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

Output:

(x*(3 - 2*x^2 - x^4)^(3/2))/7 + (6*((x*(13 - 3*x^2)*Sqrt[3 - 2*x^2 - x^4]) 
/15 + (8*(-7*Sqrt[3]*EllipticE[ArcSin[x], -1/3] + 11*Sqrt[3]*EllipticF[Arc 
Sin[x], -1/3]))/15))/7
 

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 1404

Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b 
*x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1))   Int[(2*a + b*x^2)*( 
a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* 
c, 0] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && 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 [A] (verified)

Time = 2.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.70

method	result	size
risch	\(\frac {x \left (5 x^{4}+16 x^{2}-41\right ) \left (x^{4}+2 x^{2}-3\right )}{35 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {64 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{35 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {16 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{5 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(136\)
default	\(-\frac {x^{5} \sqrt {-x^{4}-2 x^{2}+3}}{7}-\frac {16 x^{3} \sqrt {-x^{4}-2 x^{2}+3}}{35}+\frac {41 x \sqrt {-x^{4}-2 x^{2}+3}}{35}+\frac {64 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{35 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {16 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{5 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(152\)
elliptic	\(-\frac {x^{5} \sqrt {-x^{4}-2 x^{2}+3}}{7}-\frac {16 x^{3} \sqrt {-x^{4}-2 x^{2}+3}}{35}+\frac {41 x \sqrt {-x^{4}-2 x^{2}+3}}{35}+\frac {64 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{35 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {16 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{5 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(152\)

Input:

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

Output:

1/35*x*(5*x^4+16*x^2-41)*(x^4+2*x^2-3)/(-x^4-2*x^2+3)^(1/2)+64/35*(-x^2+1) 
^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/2)*EllipticF(x,1/3*I*3^(1/2))+16/ 
5*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/2)*(EllipticF(x,1/3*I*3 
^(1/2))-EllipticE(x,1/3*I*3^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \left (3-2 x^2-x^4\right )^{3/2} \, dx=\frac {112 i \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-3) + 80 i \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-3) - {\left (5 \, x^{6} + 16 \, x^{4} - 41 \, x^{2} - 112\right )} \sqrt {-x^{4} - 2 \, x^{2} + 3}}{35 \, x} \] Input:

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

Output:

1/35*(112*I*x*elliptic_e(arcsin(1/x), -3) + 80*I*x*elliptic_f(arcsin(1/x), 
 -3) - (5*x^6 + 16*x^4 - 41*x^2 - 112)*sqrt(-x^4 - 2*x^2 + 3))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (3-2 x^2-x^4\right )^{3/2} \, dx=-\frac {\sqrt {-x^{4}-2 x^{2}+3}\, x^{5}}{7}-\frac {16 \sqrt {-x^{4}-2 x^{2}+3}\, x^{3}}{35}+\frac {41 \sqrt {-x^{4}-2 x^{2}+3}\, x}{35}-\frac {192 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{4}+2 x^{2}-3}d x \right )}{35}+\frac {16 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}\, x^{2}}{x^{4}+2 x^{2}-3}d x \right )}{5} \] Input:

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

Output:

( - 5*sqrt( - x**4 - 2*x**2 + 3)*x**5 - 16*sqrt( - x**4 - 2*x**2 + 3)*x**3 
 + 41*sqrt( - x**4 - 2*x**2 + 3)*x - 192*int(sqrt( - x**4 - 2*x**2 + 3)/(x 
**4 + 2*x**2 - 3),x) + 112*int((sqrt( - x**4 - 2*x**2 + 3)*x**2)/(x**4 + 2 
*x**2 - 3),x))/35