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

Optimal. Leaf size=169 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{a+b x^2-c x^4}} \]

[Out]

(Sqrt[b + Sqrt[b^2 + 4*a*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])]*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])

_______________________________________________________________________________________

Rubi [A]  time = 0.196483, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b + Sqrt[b^2 + 4*a*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])]*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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.7662, size = 151, normalized size = 0.89 \[ \frac{\sqrt{2} \sqrt{b + \sqrt{4 a c + b^{2}}} \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{2 \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.141634, size = 177, normalized size = 1.05 \[ -\frac{i \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )}{\sqrt{2} \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 145, normalized size = 0.9 \[{\frac{\sqrt{2}}{4}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-c*x^4+b*x^2+a)^(1/2),x)

[Out]

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))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-c x^{4} + b x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x^{2} - c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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