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

Optimal. Leaf size=445 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \left (-b \sqrt{4 a c+b^2}+a c+b^2\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{b \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{2 b \sqrt{a+b x^2-c x^4}}{3 a^2 x}-\frac{\sqrt{a+b x^2-c x^4}}{3 a x^3} \]

[Out]

-Sqrt[a + b*x^2 - c*x^4]/(3*a*x^3) + (2*b*Sqrt[a + b*x^2 - c*x^4])/(3*a^2*x) - (
b*(b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 - (2*c*x^2)/(b - Sq
rt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(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 - Sq
rt[b^2 + 4*a*c])])/(3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + S
qrt[b^2 + 4*a*c]]*(b^2 + a*c - 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[(Sq
rt[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])])/(3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.33006, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \left (-b \sqrt{4 a c+b^2}+a c+b^2\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{b \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{2 b \sqrt{a+b x^2-c x^4}}{3 a^2 x}-\frac{\sqrt{a+b x^2-c x^4}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x^2 - c*x^4]/(3*a*x^3) + (2*b*Sqrt[a + b*x^2 - c*x^4])/(3*a^2*x) - (
b*(b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 - (2*c*x^2)/(b - Sq
rt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(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 - Sq
rt[b^2 + 4*a*c])])/(3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + S
qrt[b^2 + 4*a*c]]*(b^2 + a*c - 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[(Sq
rt[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])])/(3*Sqrt[2]*a^2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 158.343, size = 393, normalized size = 0.88 \[ - \frac{\sqrt{a + b x^{2} - c x^{4}}}{3 a x^{3}} + \frac{2 b \sqrt{a + b x^{2} - c x^{4}}}{3 a^{2} x} - \frac{\sqrt{2} b \left (b - \sqrt{4 a c + b^{2}}\right ) \sqrt{b + \sqrt{4 a c + b^{2}}} \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{6 a^{2} \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} + \frac{\sqrt{2} \sqrt{b + \sqrt{4 a c + b^{2}}} \left (a c + b \left (b - \sqrt{4 a c + b^{2}}\right )\right ) \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{6 a^{2} \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**2 - c*x**4)/(3*a*x**3) + 2*b*sqrt(a + b*x**2 - c*x**4)/(3*a**2*x)
- sqrt(2)*b*(b - sqrt(4*a*c + b**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)*ell
iptic_e(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)))/(6*a**2*sqrt(c)*sqrt(a + b*x**2 - c*x**4)) + sq
rt(2)*sqrt(b + sqrt(4*a*c + b**2))*(a*c + b*(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)))/(6*a**2*sqrt(c)*sqrt(a + b*x**2 - c*x**4))

_______________________________________________________________________________________

Mathematica [C]  time = 1.40933, size = 472, normalized size = 1.06 \[ \frac{-2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left (a-2 b x^2\right ) \left (a+b x^2-c x^4\right )+i \sqrt{2} x^3 \left (b \sqrt{4 a c+b^2}-a c-b^2\right ) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )-i \sqrt{2} b x^3 \left (\sqrt{4 a c+b^2}-b\right ) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{6 a^2 x^3 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*(a - 2*b*x^2)*(a + b*x^2 - c*x^4) - I*Sqr
t[2]*b*(-b + Sqrt[b^2 + 4*a*c])*x^3*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2)/(b +
Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 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])] + I*Sqrt[2]*(-b^2 - a*c + b*Sqrt[b^2 +
4*a*c])*x^3*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqr
t[(-b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*EllipticF[I*ArcSi
nh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - S
qrt[b^2 + 4*a*c])])/(6*a^2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x^3*Sqrt[a + b*x^2
- c*x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 417, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{2\,b}{3\,{a}^{2}x}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{c\sqrt{2}}{12\,a}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}}-{\frac{bc\sqrt{2}}{3\,a}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-c*x^4+b*x^2+a)^(1/2)/a/x^3+2/3*b*(-c*x^4+b*x^2+a)^(1/2)/a^2/x+1/12*c/a*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))-1/3*b*c/a*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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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