\(\int \frac {1}{(-3-7 x^2+2 x^4)^{3/2}} \, dx\) [231]

Optimal result

Integrand size = 16, antiderivative size = 235 \[ \int \frac {1}{\left (-3-7 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (61-14 x^2\right )}{219 \sqrt {-3-7 x^2+2 x^4}}-\frac {7 \sqrt {\frac {1}{6} \left (-7+\sqrt {73}\right )} \sqrt {6+\left (7-\sqrt {73}\right ) x^2} \sqrt {6+\left (7+\sqrt {73}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{6} \left (-7+\sqrt {73}\right )} x\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{438 \sqrt {-3-7 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{438} \left (-7+\sqrt {73}\right )} \sqrt {6+\left (7-\sqrt {73}\right ) x^2} \sqrt {6+\left (7+\sqrt {73}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{6} \left (-7+\sqrt {73}\right )} x\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{6 \sqrt {-3-7 x^2+2 x^4}} \] Output:

-1/219*x*(-14*x^2+61)/(2*x^4-7*x^2-3)^(1/2)-7/2628*(-42+6*73^(1/2))^(1/2)* 
(6+(-73^(1/2)+7)*x^2)^(1/2)*(6+(7+73^(1/2))*x^2)^(1/2)*EllipticE(1/6*(-42+ 
6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2))/(2*x^4-7*x^2-3)^(1/2) 
-1/2628*(-3066+438*73^(1/2))^(1/2)*(6+(-73^(1/2)+7)*x^2)^(1/2)*(6+(7+73^(1 
/2))*x^2)^(1/2)*EllipticF(1/6*(-42+6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12 
*I*438^(1/2))/(2*x^4-7*x^2-3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.79 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-3-7 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (-61+14 x^2\right )-14 i \sqrt {7+\sqrt {73}} \sqrt {6+14 x^2-4 x^4} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )+\frac {2 i \left (73+7 \sqrt {73}\right ) \sqrt {6+14 x^2-4 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{\sqrt {7+\sqrt {73}}}}{876 \sqrt {-3-7 x^2+2 x^4}} \] Input:

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

Output:

(4*x*(-61 + 14*x^2) - (14*I)*Sqrt[7 + Sqrt[73]]*Sqrt[6 + 14*x^2 - 4*x^4]*E 
llipticE[I*ArcSinh[(2*x)/Sqrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12] + (( 
2*I)*(73 + 7*Sqrt[73])*Sqrt[6 + 14*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/ 
Sqrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12])/Sqrt[7 + Sqrt[73]])/(876*Sqr 
t[-3 - 7*x^2 + 2*x^4])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.63, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1501, 25, 1411, 1498}

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

\(\displaystyle \frac {1}{219} \int -\frac {2 \left (7 x^2+6\right )}{\sqrt {2 x^4-7 x^2-3}}dx-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{219} \int \frac {7 x^2+6}{\sqrt {2 x^4-7 x^2-3}}dx-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 1501

\(\displaystyle -\frac {2}{219} \left (\frac {1}{4} \left (73+7 \sqrt {73}\right ) \int \frac {1}{\sqrt {2 x^4-7 x^2-3}}dx+\frac {7}{4} \int -\frac {-4 x^2+\sqrt {73}+7}{\sqrt {2 x^4-7 x^2-3}}dx\right )-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2}{219} \left (\frac {1}{4} \left (73+7 \sqrt {73}\right ) \int \frac {1}{\sqrt {2 x^4-7 x^2-3}}dx-\frac {7}{4} \int \frac {-4 x^2+\sqrt {73}+7}{\sqrt {2 x^4-7 x^2-3}}dx\right )-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 1411

\(\displaystyle -\frac {2}{219} \left (\frac {\left (73+7 \sqrt {73}\right ) \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+6}{\left (7-\sqrt {73}\right ) x^2+6}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6}}\right ),\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{8 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+6}} \sqrt {2 x^4-7 x^2-3}}-\frac {7}{4} \int \frac {-4 x^2+\sqrt {73}+7}{\sqrt {2 x^4-7 x^2-3}}dx\right )-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

\(\Big \downarrow \) 1498

\(\displaystyle -\frac {2}{219} \left (\frac {\left (73+7 \sqrt {73}\right ) \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+6}{\left (7-\sqrt {73}\right ) x^2+6}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6}}\right ),\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{8 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+6}} \sqrt {2 x^4-7 x^2-3}}-\frac {7}{4} \left (\frac {\sqrt [4]{73} \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+6}{\left (7-\sqrt {73}\right ) x^2+6}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-6}}\right )|\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{\sqrt {3} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+6}} \sqrt {2 x^4-7 x^2-3}}+\frac {x \left (-4 x^2-\sqrt {73}+7\right )}{\sqrt {2 x^4-7 x^2-3}}\right )\right )-\frac {x \left (61-14 x^2\right )}{219 \sqrt {2 x^4-7 x^2-3}}\)

Input:

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

Output:

-1/219*(x*(61 - 14*x^2))/Sqrt[-3 - 7*x^2 + 2*x^4] - (2*((-7*((x*(7 - Sqrt[ 
73] - 4*x^2))/Sqrt[-3 - 7*x^2 + 2*x^4] + (73^(1/4)*Sqrt[-6 - (7 - Sqrt[73] 
)*x^2]*Sqrt[(6 + (7 + Sqrt[73])*x^2)/(6 + (7 - Sqrt[73])*x^2)]*EllipticE[A 
rcSin[(Sqrt[2]*73^(1/4)*x)/Sqrt[-6 - (7 - Sqrt[73])*x^2]], (73 - 7*Sqrt[73 
])/146])/(Sqrt[3]*Sqrt[(6 + (7 - Sqrt[73])*x^2)^(-1)]*Sqrt[-3 - 7*x^2 + 2* 
x^4])))/4 + ((73 + 7*Sqrt[73])*Sqrt[-6 - (7 - Sqrt[73])*x^2]*Sqrt[(6 + (7 
+ Sqrt[73])*x^2)/(6 + (7 - Sqrt[73])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*73^(1 
/4)*x)/Sqrt[-6 - (7 - Sqrt[73])*x^2]], (73 - 7*Sqrt[73])/146])/(8*Sqrt[3]* 
73^(1/4)*Sqrt[(6 + (7 - Sqrt[73])*x^2)^(-1)]*Sqrt[-3 - 7*x^2 + 2*x^4])))/2 
19
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1411

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[( 
2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2) 
]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] 
] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

rule 1498

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[e*x*((b + q + 2*c*x^2)/(2*c*Sqrt[ 
a + b*x^2 + c*x^4])), x] - Simp[e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q) 
*x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2* 
a + (b + q)*x^2)]))*EllipticE[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b 
 + q)/(2*q)], x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] 
 && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

rule 1501

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*c*d - e*(b - q))/(2*c)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[e/(2*c)   Int[(b - q + 2*c*x^2)/Sqr 
t[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, 
c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {x \left (14 x^{2}-61\right )}{219 \sqrt {2 x^{4}-7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}\, \left (-7+\sqrt {73}\right )}\)	\(228\)
default	\(-\frac {4 \left (\frac {61}{876} x -\frac {7}{438} x^{3}\right )}{\sqrt {2 x^{4}-7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}\, \left (-7+\sqrt {73}\right )}\)	\(229\)
elliptic	\(-\frac {4 \left (\frac {61}{876} x -\frac {7}{438} x^{3}\right )}{\sqrt {2 x^{4}-7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42-6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-42-6 \sqrt {73}}\, \sqrt {2 x^{4}-7 x^{2}-3}\, \left (-7+\sqrt {73}\right )}\)	\(229\)

Input:

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

Output:

1/219*x*(14*x^2-61)/(2*x^4-7*x^2-3)^(1/2)-24/73/(-42-6*73^(1/2))^(1/2)*(1- 
(-7/6-1/6*73^(1/2))*x^2)^(1/2)*(1-(-7/6+1/6*73^(1/2))*x^2)^(1/2)/(2*x^4-7* 
x^2-3)^(1/2)*EllipticF(1/6*(-42-6*73^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12* 
I*6^(1/2))-168/73/(-42-6*73^(1/2))^(1/2)*(1-(-7/6-1/6*73^(1/2))*x^2)^(1/2) 
*(1-(-7/6+1/6*73^(1/2))*x^2)^(1/2)/(2*x^4-7*x^2-3)^(1/2)/(-7+73^(1/2))*(El 
lipticF(1/6*(-42-6*73^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12*I*6^(1/2))-Elli 
pticE(1/6*(-42-6*73^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (-3-7 x^2+2 x^4\right )^{3/2}} \, dx=\frac {7 \, {\left (\sqrt {73} \sqrt {-3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )} - 7 \, \sqrt {-3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}} E(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) - {\left (\sqrt {73} \sqrt {-3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )} - 91 \, \sqrt {-3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}} F(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) + 6 \, \sqrt {2 \, x^{4} - 7 \, x^{2} - 3} {\left (14 \, x^{3} - 61 \, x\right )}}{1314 \, {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}} \] Input:

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

Output:

1/1314*(7*(sqrt(73)*sqrt(-3)*(2*x^4 - 7*x^2 - 3) - 7*sqrt(-3)*(2*x^4 - 7*x 
^2 - 3))*sqrt(1/6*sqrt(73) - 7/6)*elliptic_e(arcsin(x*sqrt(1/6*sqrt(73) - 
7/6)), -7/12*sqrt(73) - 61/12) - (sqrt(73)*sqrt(-3)*(2*x^4 - 7*x^2 - 3) - 
91*sqrt(-3)*(2*x^4 - 7*x^2 - 3))*sqrt(1/6*sqrt(73) - 7/6)*elliptic_f(arcsi 
n(x*sqrt(1/6*sqrt(73) - 7/6)), -7/12*sqrt(73) - 61/12) + 6*sqrt(2*x^4 - 7* 
x^2 - 3)*(14*x^3 - 61*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/(2*x^4 - 7*x^2 - 3)^(3/2),x)
 

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (-3-7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}-7 x^{2}-3}}{4 x^{8}-28 x^{6}+37 x^{4}+42 x^{2}+9}d x \] 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 + 37*x**4 + 42*x**2 + 9),x 
)