\(\int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx\) [1006]

Optimal result

Integrand size = 21, antiderivative size = 408 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {a+b x^2-c x^4}}{a x}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}} \] Output:

-(-c*x^4+b*x^2+a)^(1/2)/a/x+1/4*(b-(4*a*c+b^2)^(1/2))*(b+(4*a*c+b^2)^(1/2) 
)^(1/2)*(1-2*c*x^2/(b-(4*a*c+b^2)^(1/2)))^(1/2)*(1-2*c*x^2/(b+(4*a*c+b^2)^ 
(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*x/(b+(4*a*c+b^2)^(1/2))^(1/2),((b+ 
(4*a*c+b^2)^(1/2))/(b-(4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/a/c^(1/2)/(-c*x^4 
+b*x^2+a)^(1/2)-1/4*(b-(4*a*c+b^2)^(1/2))*(b+(4*a*c+b^2)^(1/2))^(1/2)*(1-2 
*c*x^2/(b-(4*a*c+b^2)^(1/2)))^(1/2)*(1-2*c*x^2/(b+(4*a*c+b^2)^(1/2)))^(1/2 
)*EllipticF(2^(1/2)*c^(1/2)*x/(b+(4*a*c+b^2)^(1/2))^(1/2),((b+(4*a*c+b^2)^ 
(1/2))/(b-(4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/a/c^(1/2)/(-c*x^4+b*x^2+a)^(1 
/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\frac {-\frac {4 a}{x}-4 b x+4 c x^3+\frac {i \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {2+\frac {4 c x^2}{-b+\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )\right )}{\sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}}}}{4 a \sqrt {a+b x^2-c x^4}} \] Input:

Integrate[1/(x^2*Sqrt[a + b*x^2 - c*x^4]),x]
 

Output:

((-4*a)/x - 4*b*x + 4*c*x^3 + (I*(-b + Sqrt[b^2 + 4*a*c])*Sqrt[2 + (4*c*x^ 
2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*( 
EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sq 
rt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sq 
rt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 
 + 4*a*c])]))/Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))])/(4*a*Sqrt[a + b*x^2 - c* 
x^4])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1443, 25, 27, 1460, 389, 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.

\(\displaystyle \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int -\frac {c x^2}{\sqrt {-c x^4+b x^2+a}}dx}{a}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {c x^2}{\sqrt {-c x^4+b x^2+a}}dx}{a}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c \int \frac {x^2}{\sqrt {-c x^4+b x^2+a}}dx}{a}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 1460

\(\displaystyle -\frac {c \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \int \frac {x^2}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{a \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 389

\(\displaystyle -\frac {c \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}\right )}{a \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {c \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}}dx}{2 c}\right )}{a \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {c \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2}}\right )}{a \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {a+b x^2-c x^4}}{a x}\)

Input:

Int[1/(x^2*Sqrt[a + b*x^2 - c*x^4]),x]
 

Output:

-(Sqrt[a + b*x^2 - c*x^4]/(a*x)) - (c*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4 
*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*(-1/2*((b - Sqrt[b^2 + 
 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/ 
Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a* 
c])])/(Sqrt[2]*c^(3/2)) + ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a 
*c]]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b 
 + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(2*Sqrt[2]*c^(3/2))))/(a*S 
qrt[a + b*x^2 - c*x^4])
 

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 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 389

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[1/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b   Int 
[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N 
eQ[b*c - a*d, 0] &&  !SimplerSqrtQ[-b/a, -d/c]
 

rule 1443

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim 
p[1/(a*d^2*(m + 1))   Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* 
x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 
- 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1460

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

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(241\)
risch	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(241\)
elliptic	\(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(241\)

Input:

int(1/x^2/(-c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-(-c*x^4+b*x^2+a)^(1/2)/a/x+1/2*c*2^(1/2)/((-b+(4*a*c+b^2)^(1/2))/a)^(1/2) 
*(4-2*(-b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(4*a*c+b^2)^(1/2))/a*x^2 
)^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1 
/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c 
)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4- 
2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} x \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {a} b x\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} x \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {a} b x\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} a}{2 \, a^{2} x} \] Input:

integrate(1/x^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/2*(sqrt(1/2)*(a^(3/2)*x*sqrt((b^2 + 4*a*c)/a^2) - sqrt(a)*b*x)*sqrt((a* 
sqrt((b^2 + 4*a*c)/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt 
((b^2 + 4*a*c)/a^2) - b)/a)), -1/2*(a*b*sqrt((b^2 + 4*a*c)/a^2) + b^2 + 2* 
a*c)/(a*c)) - sqrt(1/2)*(a^(3/2)*x*sqrt((b^2 + 4*a*c)/a^2) - sqrt(a)*b*x)* 
sqrt((a*sqrt((b^2 + 4*a*c)/a^2) - b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*sqrt 
((a*sqrt((b^2 + 4*a*c)/a^2) - b)/a)), -1/2*(a*b*sqrt((b^2 + 4*a*c)/a^2) + 
b^2 + 2*a*c)/(a*c)) + 2*sqrt(-c*x^4 + b*x^2 + a)*a)/(a^2*x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b x^{2} - c x^{4}}}\, dx \] Input:

integrate(1/x**2/(-c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral(1/(x**2*sqrt(a + b*x**2 - c*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{2}} \,d x } \] Input:

integrate(1/x^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*x^2), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{2}} \,d x } \] Input:

integrate(1/x^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^2\,\sqrt {-c\,x^4+b\,x^2+a}} \,d x \] Input:

int(1/(x^2*(a + b*x^2 - c*x^4)^(1/2)),x)
 

Output:

int(1/(x^2*(a + b*x^2 - c*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{-c \,x^{6}+b \,x^{4}+a \,x^{2}}d x \] Input:

int(1/x^2/(-c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int(sqrt(a + b*x**2 - c*x**4)/(a*x**2 + b*x**4 - c*x**6),x)