\(\int \frac {1}{\sqrt {a+b x^2-c x^4}} \, dx\) [1005]

Optimal result

Integrand size = 17, antiderivative size = 169 \[ \int \frac {1}{\sqrt {a+b x^2-c x^4}} \, dx=\frac {\sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {a+b x^2-c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {i \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),-\frac {b+\sqrt {b^2+4 a c}}{-b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {a+b x^2-c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1417, 321}

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.

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

\(\Big \downarrow \) 1417

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1417

Int[1/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[1/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[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 = 0.65 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\)	\(145\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\)	\(145\)

Input:

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

Output:

1/4*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) 
*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))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {a+b x^2-c x^4}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + \sqrt {a} b\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c})}{2 \, a c} \] Input:

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

Output:

1/2*sqrt(1/2)*(a^(3/2)*sqrt((b^2 + 4*a*c)/a^2) + sqrt(a)*b)*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))/(a*c)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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