\(\int \frac {1}{(-3-7 x^2-2 x^4)^{3/2}} \, dx\) [277]

Optimal result

Integrand size = 16, antiderivative size = 94 \[ \int \frac {1}{\left (-3-7 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {2 x}{5 \sqrt {-3-7 x^2-2 x^4}}+\frac {7 \sqrt {-1-2 x^2} E\left (\left .\arctan \left (\frac {x}{\sqrt {3}}\right )\right |-5\right )}{75 \sqrt {1+2 x^2}}+\frac {4 \sqrt {1+2 x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{\sqrt {3}}\right ),-5\right )}{25 \sqrt {-1-2 x^2}} \] Output:

-2/5*x/(-2*x^4-7*x^2-3)^(1/2)+7/75*(-2*x^2-1)^(1/2)*EllipticE(x*3^(1/2)/(3 
*x^2+9)^(1/2),I*5^(1/2))/(2*x^2+1)^(1/2)+4/25*(2*x^2+1)^(1/2)*InverseJacob 
iAM(arctan(1/3*x*3^(1/2)),I*5^(1/2))/(-2*x^2-1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (-3-7 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {37 x+14 x^3+7 i \sqrt {6} \sqrt {3+x^2} \sqrt {1+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {2} x\right )|\frac {1}{6}\right )-5 i \sqrt {6} \sqrt {3+x^2} \sqrt {1+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} x\right ),\frac {1}{6}\right )}{75 \sqrt {-3-7 x^2-2 x^4}} \] Input:

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

Output:

-1/75*(37*x + 14*x^3 + (7*I)*Sqrt[6]*Sqrt[3 + x^2]*Sqrt[1 + 2*x^2]*Ellipti 
cE[I*ArcSinh[Sqrt[2]*x], 1/6] - (5*I)*Sqrt[6]*Sqrt[3 + x^2]*Sqrt[1 + 2*x^2 
]*EllipticF[I*ArcSinh[Sqrt[2]*x], 1/6])/Sqrt[-3 - 7*x^2 - 2*x^4]
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.77, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1405, 27, 1494, 27, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{75} \int \frac {2 \left (7 x^2+6\right )}{\sqrt {-2 x^4-7 x^2-3}}dx-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{75} \int \frac {7 x^2+6}{\sqrt {-2 x^4-7 x^2-3}}dx-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {4}{75} \sqrt {2} \int \frac {7 x^2+6}{2 \sqrt {2} \sqrt {-2 x^2-1} \sqrt {x^2+3}}dx-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{75} \int \frac {7 x^2+6}{\sqrt {-2 x^2-1} \sqrt {x^2+3}}dx-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {2}{75} \left (6 \int \frac {1}{\sqrt {-2 x^2-1} \sqrt {x^2+3}}dx+7 \int \frac {x^2}{\sqrt {-2 x^2-1} \sqrt {x^2+3}}dx\right )-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {2}{75} \left (7 \int \frac {x^2}{\sqrt {-2 x^2-1} \sqrt {x^2+3}}dx-\frac {6 \sqrt {-2 x^2-1} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{\sqrt {3}}\right ),-5\right )}{\sqrt {x^2+3} \sqrt {\frac {2 x^2+1}{x^2+3}}}\right )-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {2}{75} \left (7 \left (\frac {3}{2} \int \frac {\sqrt {-2 x^2-1}}{\left (x^2+3\right )^{3/2}}dx-\frac {x \sqrt {-2 x^2-1}}{2 \sqrt {x^2+3}}\right )-\frac {6 \sqrt {-2 x^2-1} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{\sqrt {3}}\right ),-5\right )}{\sqrt {x^2+3} \sqrt {\frac {2 x^2+1}{x^2+3}}}\right )-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {2}{75} \left (7 \left (\frac {\sqrt {-2 x^2-1} E\left (\left .\arctan \left (\frac {x}{\sqrt {3}}\right )\right |-5\right )}{2 \sqrt {x^2+3} \sqrt {\frac {2 x^2+1}{x^2+3}}}-\frac {x \sqrt {-2 x^2-1}}{2 \sqrt {x^2+3}}\right )-\frac {6 \sqrt {-2 x^2-1} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{\sqrt {3}}\right ),-5\right )}{\sqrt {x^2+3} \sqrt {\frac {2 x^2+1}{x^2+3}}}\right )-\frac {x \left (14 x^2+37\right )}{75 \sqrt {-2 x^4-7 x^2-3}}\)

Input:

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

Output:

-1/75*(x*(37 + 14*x^2))/Sqrt[-3 - 7*x^2 - 2*x^4] + (2*(7*(-1/2*(x*Sqrt[-1 
- 2*x^2])/Sqrt[3 + x^2] + (Sqrt[-1 - 2*x^2]*EllipticE[ArcTan[x/Sqrt[3]], - 
5])/(2*Sqrt[3 + x^2]*Sqrt[(1 + 2*x^2)/(3 + x^2)])) - (6*Sqrt[-1 - 2*x^2]*E 
llipticF[ArcTan[x/Sqrt[3]], -5])/(Sqrt[3 + x^2]*Sqrt[(1 + 2*x^2)/(3 + x^2) 
])))/75
 

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

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 [A] (verified)

Time = 2.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.53

method	result	size
risch	\(-\frac {x \left (14 x^{2}+37\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}+\frac {7 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}\)	\(144\)
default	\(\frac {-\frac {37}{75} x -\frac {14}{75} x^{3}}{\sqrt {-2 x^{4}-7 x^{2}-3}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}+\frac {7 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}\)	\(145\)
elliptic	\(\frac {-\frac {37}{75} x -\frac {14}{75} x^{3}}{\sqrt {-2 x^{4}-7 x^{2}-3}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}+\frac {7 i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (i x \sqrt {2}, \frac {\sqrt {6}}{6}\right )\right )}{75 \sqrt {-2 x^{4}-7 x^{2}-3}}\)	\(145\)

Input:

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

Output:

-1/75*x*(14*x^2+37)/(-2*x^4-7*x^2-3)^(1/2)-2/75*I*2^(1/2)*(2*x^2+1)^(1/2)* 
(3*x^2+9)^(1/2)/(-2*x^4-7*x^2-3)^(1/2)*EllipticF(I*2^(1/2)*x,1/6*6^(1/2))+ 
7/75*I*2^(1/2)*(2*x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-2*x^4-7*x^2-3)^(1/2)*(Ell 
ipticF(I*2^(1/2)*x,1/6*6^(1/2))-EllipticE(I*2^(1/2)*x,1/6*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (-3-7 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {7 \, \sqrt {-\frac {1}{3}} \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )} E(\arcsin \left (\sqrt {-\frac {1}{3}} x\right )\,|\,6) - 43 \, \sqrt {-\frac {1}{3}} \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )} F(\arcsin \left (\sqrt {-\frac {1}{3}} x\right )\,|\,6) - 3 \, \sqrt {-2 \, x^{4} - 7 \, x^{2} - 3} {\left (14 \, x^{3} + 37 \, x\right )}}{225 \, {\left (2 \, x^{4} + 7 \, x^{2} + 3\right )}} \] Input:

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

Output:

-1/225*(7*sqrt(-1/3)*sqrt(-3)*(2*x^4 + 7*x^2 + 3)*elliptic_e(arcsin(sqrt(- 
1/3)*x), 6) - 43*sqrt(-1/3)*sqrt(-3)*(2*x^4 + 7*x^2 + 3)*elliptic_f(arcsin 
(sqrt(-1/3)*x), 6) - 3*sqrt(-2*x^4 - 7*x^2 - 3)*(14*x^3 + 37*x))/(2*x^4 + 
7*x^2 + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (-3-7 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-2 x^{4}-7 x^{2}-3}}{4 x^{8}+28 x^{6}+61 x^{4}+42 x^{2}+9}d x \] Input:

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

Output:

int(sqrt( - 2*x**4 - 7*x**2 - 3)/(4*x**8 + 28*x**6 + 61*x**4 + 42*x**2 + 9 
),x)