3.974 \(\int \frac{1}{x^5 \sqrt{-a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{16 a^{5/2}}+\frac{3 b \sqrt{-a+b x^2+c x^4}}{8 a^2 x^2}+\frac{\sqrt{-a+b x^2+c x^4}}{4 a x^4} \]

[Out]

Sqrt[-a + b*x^2 + c*x^4]/(4*a*x^4) + (3*b*Sqrt[-a + b*x^2 + c*x^4])/(8*a^2*x^2)
- ((3*b^2 + 4*a*c)*ArcTan[(2*a - b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(
16*a^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.269914, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{16 a^{5/2}}+\frac{3 b \sqrt{-a+b x^2+c x^4}}{8 a^2 x^2}+\frac{\sqrt{-a+b x^2+c x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

Sqrt[-a + b*x^2 + c*x^4]/(4*a*x^4) + (3*b*Sqrt[-a + b*x^2 + c*x^4])/(8*a^2*x^2)
- ((3*b^2 + 4*a*c)*ArcTan[(2*a - b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(
16*a^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.6516, size = 97, normalized size = 0.84 \[ \frac{\sqrt{- a + b x^{2} + c x^{4}}}{4 a x^{4}} + \frac{3 b \sqrt{- a + b x^{2} + c x^{4}}}{8 a^{2} x^{2}} + \frac{\left (4 a c + 3 b^{2}\right ) \operatorname{atan}{\left (\frac{- 2 a + b x^{2}}{2 \sqrt{a} \sqrt{- a + b x^{2} + c x^{4}}} \right )}}{16 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

sqrt(-a + b*x**2 + c*x**4)/(4*a*x**4) + 3*b*sqrt(-a + b*x**2 + c*x**4)/(8*a**2*x
**2) + (4*a*c + 3*b**2)*atan((-2*a + b*x**2)/(2*sqrt(a)*sqrt(-a + b*x**2 + c*x**
4)))/(16*a**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.189887, size = 118, normalized size = 1.03 \[ \frac{\left (4 a c+3 b^2\right ) \left (\frac{\log (x)}{\sqrt{-a}}-\frac{\log \left (2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-2 a+b x^2\right )}{2 \sqrt{-a}}\right )}{8 a^2}+\left (\frac{3 b}{8 a^2 x^2}+\frac{1}{4 a x^4}\right ) \sqrt{-a+b x^2+c x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

(1/(4*a*x^4) + (3*b)/(8*a^2*x^2))*Sqrt[-a + b*x^2 + c*x^4] + ((3*b^2 + 4*a*c)*(L
og[x]/Sqrt[-a] - Log[-2*a + b*x^2 + 2*Sqrt[-a]*Sqrt[-a + b*x^2 + c*x^4]]/(2*Sqrt
[-a])))/(8*a^2)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 149, normalized size = 1.3 \[{\frac{1}{4\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}-a}}+{\frac{3\,b}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{3\,{b}^{2}}{16\,{a}^{2}}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}}-{\frac{c}{4\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^4+b*x^2-a)^(1/2),x)

[Out]

1/4*(c*x^4+b*x^2-a)^(1/2)/a/x^4+3/8*b*(c*x^4+b*x^2-a)^(1/2)/a^2/x^2-3/16*b^2/a^2
/(-a)^(1/2)*ln((-2*a+b*x^2+2*(-a)^(1/2)*(c*x^4+b*x^2-a)^(1/2))/x^2)-1/4*c/a/(-a)
^(1/2)*ln((-2*a+b*x^2+2*(-a)^(1/2)*(c*x^4+b*x^2-a)^(1/2))/x^2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.30244, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b^{2} + 4 \, a c\right )} x^{4} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (a b x^{2} - 2 \, a^{2}\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{-a}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (3 \, b x^{2} + 2 \, a\right )} \sqrt{-a}}{32 \, \sqrt{-a} a^{2} x^{4}}, \frac{{\left (3 \, b^{2} + 4 \, a c\right )} x^{4} \arctan \left (\frac{b x^{2} - 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} - a}{\left (3 \, b x^{2} + 2 \, a\right )} \sqrt{a}}{16 \, a^{\frac{5}{2}} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^5),x, algorithm="fricas")

[Out]

[1/32*((3*b^2 + 4*a*c)*x^4*log((4*sqrt(c*x^4 + b*x^2 - a)*(a*b*x^2 - 2*a^2) + ((
b^2 - 4*a*c)*x^4 - 8*a*b*x^2 + 8*a^2)*sqrt(-a))/x^4) + 4*sqrt(c*x^4 + b*x^2 - a)
*(3*b*x^2 + 2*a)*sqrt(-a))/(sqrt(-a)*a^2*x^4), 1/16*((3*b^2 + 4*a*c)*x^4*arctan(
1/2*(b*x^2 - 2*a)/(sqrt(c*x^4 + b*x^2 - a)*sqrt(a))) + 2*sqrt(c*x^4 + b*x^2 - a)
*(3*b*x^2 + 2*a)*sqrt(a))/(a^(5/2)*x^4)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(-a + b*x**2 + c*x**4)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.522576, size = 122, normalized size = 1.06 \[ \frac{1}{8} \, \sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}{\left (\frac{3 \, b}{a^{2}} + \frac{2}{a x^{2}}\right )} + \frac{{\left (3 \, b^{2} + 4 \, a c\right )}{\rm ln}\left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{16 \, \sqrt{-a} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^5),x, algorithm="giac")

[Out]

1/8*sqrt(c + b/x^2 - a/x^4)*(3*b/a^2 + 2/(a*x^2)) + 1/16*(3*b^2 + 4*a*c)*ln(abs(
-2*sqrt(-a)*(sqrt(c + b/x^2 - a/x^4) - sqrt(-a)/x^2) + b))/(sqrt(-a)*a^2)