\(\int \frac {1}{(3-2 x^2-x^4)^{7/2}} \, dx\) [342]

Optimal result

Integrand size = 16, antiderivative size = 113 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\frac {x \left (5+x^2\right )}{120 \left (3-2 x^2-x^4\right )^{5/2}}+\frac {7 x \left (7+2 x^2\right )}{2160 \left (3-2 x^2-x^4\right )^{3/2}}+\frac {7 x \left (133+41 x^2\right )}{51840 \sqrt {3-2 x^2-x^4}}-\frac {287 E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )}{17280 \sqrt {3}}+\frac {203 \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )}{8640 \sqrt {3}} \] Output:

1/120*x*(x^2+5)/(-x^4-2*x^2+3)^(5/2)+7/2160*x*(2*x^2+7)/(-x^4-2*x^2+3)^(3/ 
2)+7/51840*x*(41*x^2+133)/(-x^4-2*x^2+3)^(1/2)-287/51840*EllipticE(x,1/3*I 
*3^(1/2))*3^(1/2)+203/25920*EllipticF(x,1/3*I*3^(1/2))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\frac {14067 x-9501 x^3-7154 x^5+2814 x^7+2079 x^9+287 x^{11}-287 i \left (3-2 x^2-x^4\right )^{5/2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-70 i \left (3-2 x^2-x^4\right )^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )}{51840 \left (3-2 x^2-x^4\right )^{5/2}} \] Input:

Integrate[(3 - 2*x^2 - x^4)^(-7/2),x]
 

Output:

(14067*x - 9501*x^3 - 7154*x^5 + 2814*x^7 + 2079*x^9 + 287*x^11 - (287*I)* 
(3 - 2*x^2 - x^4)^(5/2)*EllipticE[I*ArcSinh[x/Sqrt[3]], -3] - (70*I)*(3 - 
2*x^2 - x^4)^(5/2)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3])/(51840*(3 - 2*x^2 
- x^4)^(5/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1405, 27, 1492, 27, 1492, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}-\frac {1}{240} \int -\frac {14 \left (x^2+5\right )}{\left (-x^4-2 x^2+3\right )^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{120} \int \frac {x^2+5}{\left (-x^4-2 x^2+3\right )^{5/2}}dx+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {7}{120} \left (\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}-\frac {1}{144} \int -\frac {8 \left (6 x^2+23\right )}{\left (-x^4-2 x^2+3\right )^{3/2}}dx\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \int \frac {6 x^2+23}{\left (-x^4-2 x^2+3\right )^{3/2}}dx+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}-\frac {1}{48} \int -\frac {2 \left (51-41 x^2\right )}{\sqrt {-x^4-2 x^2+3}}dx\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{24} \int \frac {51-41 x^2}{\sqrt {-x^4-2 x^2+3}}dx+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{12} \int \frac {51-41 x^2}{2 \sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{24} \int \frac {51-41 x^2}{\sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{24} \left (174 \int \frac {1}{\sqrt {1-x^2} \sqrt {x^2+3}}dx-41 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{24} \left (58 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-41 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {7}{120} \left (\frac {1}{18} \left (\frac {1}{24} \left (58 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-41 \sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )\right )+\frac {x \left (41 x^2+133\right )}{24 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (2 x^2+7\right )}{18 \left (-x^4-2 x^2+3\right )^{3/2}}\right )+\frac {x \left (x^2+5\right )}{120 \left (-x^4-2 x^2+3\right )^{5/2}}\)

Input:

Int[(3 - 2*x^2 - x^4)^(-7/2),x]
 

Output:

(x*(5 + x^2))/(120*(3 - 2*x^2 - x^4)^(5/2)) + (7*((x*(7 + 2*x^2))/(18*(3 - 
 2*x^2 - x^4)^(3/2)) + ((x*(133 + 41*x^2))/(24*Sqrt[3 - 2*x^2 - x^4]) + (- 
41*Sqrt[3]*EllipticE[ArcSin[x], -1/3] + 58*Sqrt[3]*EllipticF[ArcSin[x], -1 
/3])/24)/18))/120
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*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] && 
 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 [A] (verified)

Time = 2.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.35

method	result	size
risch	\(\frac {x \left (287 x^{10}+2079 x^{8}+2814 x^{6}-7154 x^{4}-9501 x^{2}+14067\right )}{51840 \left (x^{4}+2 x^{2}-3\right )^{2} \sqrt {-x^{4}-2 x^{2}+3}}+\frac {119 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {287 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(153\)
default	\(\frac {\left (-\frac {1}{24} x -\frac {1}{120} x^{3}\right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{3}}+\frac {\left (\frac {7}{1080} x^{3}+\frac {49}{2160} x \right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{2}}+\frac {\frac {287}{51840} x^{3}+\frac {931}{51840} x}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {119 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {287 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(194\)
elliptic	\(\frac {\left (-\frac {1}{24} x -\frac {1}{120} x^{3}\right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{3}}+\frac {\left (\frac {7}{1080} x^{3}+\frac {49}{2160} x \right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{2}}+\frac {\frac {287}{51840} x^{3}+\frac {931}{51840} x}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {119 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {287 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{51840 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(194\)

Input:

int(1/(-x^4-2*x^2+3)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/51840*x*(287*x^10+2079*x^8+2814*x^6-7154*x^4-9501*x^2+14067)/(x^4+2*x^2- 
3)^2/(-x^4-2*x^2+3)^(1/2)+119/51840*(-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))+287/51840*(-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 = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=-\frac {287 \, \sqrt {3} {\left (x^{12} + 6 \, x^{10} + 3 \, x^{8} - 28 \, x^{6} - 9 \, x^{4} + 54 \, x^{2} - 27\right )} E(\arcsin \left (x\right )\,|\,-\frac {1}{3}) - 406 \, \sqrt {3} {\left (x^{12} + 6 \, x^{10} + 3 \, x^{8} - 28 \, x^{6} - 9 \, x^{4} + 54 \, x^{2} - 27\right )} F(\arcsin \left (x\right )\,|\,-\frac {1}{3}) + {\left (287 \, x^{11} + 2079 \, x^{9} + 2814 \, x^{7} - 7154 \, x^{5} - 9501 \, x^{3} + 14067 \, x\right )} \sqrt {-x^{4} - 2 \, x^{2} + 3}}{51840 \, {\left (x^{12} + 6 \, x^{10} + 3 \, x^{8} - 28 \, x^{6} - 9 \, x^{4} + 54 \, x^{2} - 27\right )}} \] Input:

integrate(1/(-x^4-2*x^2+3)^(7/2),x, algorithm="fricas")
 

Output:

-1/51840*(287*sqrt(3)*(x^12 + 6*x^10 + 3*x^8 - 28*x^6 - 9*x^4 + 54*x^2 - 2 
7)*elliptic_e(arcsin(x), -1/3) - 406*sqrt(3)*(x^12 + 6*x^10 + 3*x^8 - 28*x 
^6 - 9*x^4 + 54*x^2 - 27)*elliptic_f(arcsin(x), -1/3) + (287*x^11 + 2079*x 
^9 + 2814*x^7 - 7154*x^5 - 9501*x^3 + 14067*x)*sqrt(-x^4 - 2*x^2 + 3))/(x^ 
12 + 6*x^10 + 3*x^8 - 28*x^6 - 9*x^4 + 54*x^2 - 27)
 

Sympy [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\int \frac {1}{\left (- x^{4} - 2 x^{2} + 3\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(-x**4-2*x**2+3)**(7/2),x)
 

Output:

Integral((-x**4 - 2*x**2 + 3)**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(-x^4-2*x^2+3)^(7/2),x, algorithm="maxima")
 

Output:

integrate((-x^4 - 2*x^2 + 3)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(-x^4-2*x^2+3)^(7/2),x, algorithm="giac")
 

Output:

integrate((-x^4 - 2*x^2 + 3)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\int \frac {1}{{\left (-x^4-2\,x^2+3\right )}^{7/2}} \,d x \] Input:

int(1/(3 - x^4 - 2*x^2)^(7/2),x)
 

Output:

int(1/(3 - x^4 - 2*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{7/2}} \, dx=\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{16}+8 x^{14}+12 x^{12}-40 x^{10}-74 x^{8}+120 x^{6}+108 x^{4}-216 x^{2}+81}d x \] Input:

int(1/(-x^4-2*x^2+3)^(7/2),x)
 

Output:

int(sqrt( - x**4 - 2*x**2 + 3)/(x**16 + 8*x**14 + 12*x**12 - 40*x**10 - 74 
*x**8 + 120*x**6 + 108*x**4 - 216*x**2 + 81),x)