\(\int \frac {1}{(-2-7 x^2+3 x^4)^{3/2}} \, dx\) [216]

Optimal result

Integrand size = 16, antiderivative size = 229 \[ \int \frac {1}{\left (-2-7 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (61-21 x^2\right )}{146 \sqrt {-2-7 x^2+3 x^4}}-\frac {7 \sqrt {-7+\sqrt {73}} \sqrt {4+\left (7-\sqrt {73}\right ) x^2} \sqrt {4+\left (7+\sqrt {73}\right ) x^2} E\left (\arcsin \left (\frac {1}{2} \sqrt {-7+\sqrt {73}} x\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{584 \sqrt {-2-7 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{73} \left (-7+\sqrt {73}\right )} \sqrt {4+\left (7-\sqrt {73}\right ) x^2} \sqrt {4+\left (7+\sqrt {73}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-7+\sqrt {73}} x\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{8 \sqrt {-2-7 x^2+3 x^4}} \] Output:

-1/146*x*(-21*x^2+61)/(3*x^4-7*x^2-2)^(1/2)-7/584*(-7+73^(1/2))^(1/2)*(4+( 
-73^(1/2)+7)*x^2)^(1/2)*(4+(7+73^(1/2))*x^2)^(1/2)*EllipticE(1/2*(-7+73^(1 
/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2))/(3*x^4-7*x^2-2)^(1/2)-1/584* 
(-511+73*73^(1/2))^(1/2)*(4+(-73^(1/2)+7)*x^2)^(1/2)*(4+(7+73^(1/2))*x^2)^ 
(1/2)*EllipticF(1/2*(-7+73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2)) 
/(3*x^4-7*x^2-2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.94 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (-2-7 x^2+3 x^4\right )^{3/2}} \, dx=\frac {12 x \left (-61+21 x^2\right )-42 i \sqrt {7+\sqrt {73}} \sqrt {4+14 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-7+\sqrt {73}}} x\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )+\frac {6 i \left (73+7 \sqrt {73}\right ) \sqrt {4+14 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-7+\sqrt {73}}} x\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{\sqrt {7+\sqrt {73}}}}{1752 \sqrt {-2-7 x^2+3 x^4}} \] Input:

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

Output:

(12*x*(-61 + 21*x^2) - (42*I)*Sqrt[7 + Sqrt[73]]*Sqrt[4 + 14*x^2 - 6*x^4]* 
EllipticE[I*ArcSinh[Sqrt[6/(-7 + Sqrt[73])]*x], (-61 + 7*Sqrt[73])/12] + ( 
(6*I)*(73 + 7*Sqrt[73])*Sqrt[4 + 14*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[ 
6/(-7 + Sqrt[73])]*x], (-61 + 7*Sqrt[73])/12])/Sqrt[7 + Sqrt[73]])/(1752*S 
qrt[-2 - 7*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.67, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{146} \int -\frac {3 \left (7 x^2+4\right )}{\sqrt {3 x^4-7 x^2-2}}dx-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{146} \int \frac {7 x^2+4}{\sqrt {3 x^4-7 x^2-2}}dx-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

\(\Big \downarrow \) 1501

\(\displaystyle -\frac {3}{146} \left (\frac {1}{6} \left (73+7 \sqrt {73}\right ) \int \frac {1}{\sqrt {3 x^4-7 x^2-2}}dx+\frac {7}{6} \int -\frac {-6 x^2+\sqrt {73}+7}{\sqrt {3 x^4-7 x^2-2}}dx\right )-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3}{146} \left (\frac {1}{6} \left (73+7 \sqrt {73}\right ) \int \frac {1}{\sqrt {3 x^4-7 x^2-2}}dx-\frac {7}{6} \int \frac {-6 x^2+\sqrt {73}+7}{\sqrt {3 x^4-7 x^2-2}}dx\right )-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

\(\Big \downarrow \) 1411

\(\displaystyle -\frac {3}{146} \left (\frac {\left (73+7 \sqrt {73}\right ) \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+4}{\left (7-\sqrt {73}\right ) x^2+4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4}}\right ),\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{12 \sqrt {2} \sqrt [4]{73} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+4}} \sqrt {3 x^4-7 x^2-2}}-\frac {7}{6} \int \frac {-6 x^2+\sqrt {73}+7}{\sqrt {3 x^4-7 x^2-2}}dx\right )-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

\(\Big \downarrow \) 1498

\(\displaystyle -\frac {3}{146} \left (\frac {\left (73+7 \sqrt {73}\right ) \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+4}{\left (7-\sqrt {73}\right ) x^2+4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4}}\right ),\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{12 \sqrt {2} \sqrt [4]{73} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+4}} \sqrt {3 x^4-7 x^2-2}}-\frac {7}{6} \left (\frac {\sqrt [4]{73} \sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4} \sqrt {\frac {\left (7+\sqrt {73}\right ) x^2+4}{\left (7-\sqrt {73}\right ) x^2+4}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-\left (\left (7-\sqrt {73}\right ) x^2\right )-4}}\right )|\frac {1}{146} \left (73-7 \sqrt {73}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{\left (7-\sqrt {73}\right ) x^2+4}} \sqrt {3 x^4-7 x^2-2}}+\frac {x \left (-6 x^2-\sqrt {73}+7\right )}{\sqrt {3 x^4-7 x^2-2}}\right )\right )-\frac {x \left (61-21 x^2\right )}{146 \sqrt {3 x^4-7 x^2-2}}\)

Input:

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

Output:

-1/146*(x*(61 - 21*x^2))/Sqrt[-2 - 7*x^2 + 3*x^4] - (3*((-7*((x*(7 - Sqrt[ 
73] - 6*x^2))/Sqrt[-2 - 7*x^2 + 3*x^4] + (73^(1/4)*Sqrt[-4 - (7 - Sqrt[73] 
)*x^2]*Sqrt[(4 + (7 + Sqrt[73])*x^2)/(4 + (7 - Sqrt[73])*x^2)]*EllipticE[A 
rcSin[(Sqrt[2]*73^(1/4)*x)/Sqrt[-4 - (7 - Sqrt[73])*x^2]], (73 - 7*Sqrt[73 
])/146])/(Sqrt[2]*Sqrt[(4 + (7 - Sqrt[73])*x^2)^(-1)]*Sqrt[-2 - 7*x^2 + 3* 
x^4])))/6 + ((73 + 7*Sqrt[73])*Sqrt[-4 - (7 - Sqrt[73])*x^2]*Sqrt[(4 + (7 
+ Sqrt[73])*x^2)/(4 + (7 - Sqrt[73])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*73^(1 
/4)*x)/Sqrt[-4 - (7 - Sqrt[73])*x^2]], (73 - 7*Sqrt[73])/146])/(12*Sqrt[2] 
*73^(1/4)*Sqrt[(4 + (7 - Sqrt[73])*x^2)^(-1)]*Sqrt[-2 - 7*x^2 + 3*x^4])))/ 
146
 

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 = 2.84 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {x \left (21 x^{2}-61\right )}{146 \sqrt {3 x^{4}-7 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}}-\frac {84 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}\, \left (-7+\sqrt {73}\right )}\)	\(228\)
default	\(-\frac {6 \left (\frac {61}{876} x -\frac {7}{292} x^{3}\right )}{\sqrt {3 x^{4}-7 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}}-\frac {84 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}\, \left (-7+\sqrt {73}\right )}\)	\(229\)
elliptic	\(-\frac {6 \left (\frac {61}{876} x -\frac {7}{292} x^{3}\right )}{\sqrt {3 x^{4}-7 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}}-\frac {84 \sqrt {1-\left (-\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-7-\sqrt {73}}}{2}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )\right )}{73 \sqrt {-7-\sqrt {73}}\, \sqrt {3 x^{4}-7 x^{2}-2}\, \left (-7+\sqrt {73}\right )}\)	\(229\)

Input:

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

Output:

1/146*x*(21*x^2-61)/(3*x^4-7*x^2-2)^(1/2)-12/73/(-7-73^(1/2))^(1/2)*(1-(-7 
/4-1/4*73^(1/2))*x^2)^(1/2)*(1-(-7/4+1/4*73^(1/2))*x^2)^(1/2)/(3*x^4-7*x^2 
-2)^(1/2)*EllipticF(1/2*x*(-7-73^(1/2))^(1/2),1/12*I*438^(1/2)-7/12*I*6^(1 
/2))-84/73/(-7-73^(1/2))^(1/2)*(1-(-7/4-1/4*73^(1/2))*x^2)^(1/2)*(1-(-7/4+ 
1/4*73^(1/2))*x^2)^(1/2)/(3*x^4-7*x^2-2)^(1/2)/(-7+73^(1/2))*(EllipticF(1/ 
2*x*(-7-73^(1/2))^(1/2),1/12*I*438^(1/2)-7/12*I*6^(1/2))-EllipticE(1/2*x*( 
-7-73^(1/2))^(1/2),1/12*I*438^(1/2)-7/12*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (-2-7 x^2+3 x^4\right )^{3/2}} \, dx=\frac {7 \, {\left (\sqrt {73} \sqrt {-2} {\left (3 \, x^{4} - 7 \, x^{2} - 2\right )} - 7 \, \sqrt {-2} {\left (3 \, x^{4} - 7 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} - 7} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} - 7}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) - {\left (3 \, \sqrt {73} \sqrt {-2} {\left (3 \, x^{4} - 7 \, x^{2} - 2\right )} - 77 \, \sqrt {-2} {\left (3 \, x^{4} - 7 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} - 7} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} - 7}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) + 8 \, \sqrt {3 \, x^{4} - 7 \, x^{2} - 2} {\left (21 \, x^{3} - 61 \, x\right )}}{1168 \, {\left (3 \, x^{4} - 7 \, x^{2} - 2\right )}} \] Input:

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

Output:

1/1168*(7*(sqrt(73)*sqrt(-2)*(3*x^4 - 7*x^2 - 2) - 7*sqrt(-2)*(3*x^4 - 7*x 
^2 - 2))*sqrt(sqrt(73) - 7)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(73) - 7)), - 
7/12*sqrt(73) - 61/12) - (3*sqrt(73)*sqrt(-2)*(3*x^4 - 7*x^2 - 2) - 77*sqr 
t(-2)*(3*x^4 - 7*x^2 - 2))*sqrt(sqrt(73) - 7)*elliptic_f(arcsin(1/2*x*sqrt 
(sqrt(73) - 7)), -7/12*sqrt(73) - 61/12) + 8*sqrt(3*x^4 - 7*x^2 - 2)*(21*x 
^3 - 61*x))/(3*x^4 - 7*x^2 - 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(3*x**4 - 7*x**2 - 2)/(9*x**8 - 42*x**6 + 37*x**4 + 28*x**2 + 4),x 
)