3.973 \(\int \frac{1}{x^3 \sqrt{-a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=77 \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.141881, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.1498, size = 61, normalized size = 0.79 \[ \frac{\sqrt{- a + b x^{2} + c x^{4}}}{2 a x^{2}} + \frac{b \operatorname{atan}{\left (\frac{- 2 a + b x^{2}}{2 \sqrt{a} \sqrt{- a + b x^{2} + c x^{4}}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.118517, size = 93, normalized size = 1.21 \[ -\frac{2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-b x^2 \log \left (2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-2 a+b x^2\right )+2 b x^2 \log (x)}{4 (-a)^{3/2} x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.019, size = 74, normalized size = 1. \[{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{b}{4\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^4+b*x^2-a)^(1/2),x)

[Out]

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

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293889, size = 1, normalized size = 0.01 \[ \left [\frac{b x^{2} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (a b x^{2} - 2 \, a^{2}\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{-a}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{-a}}{8 \, \sqrt{-a} a x^{2}}, \frac{b x^{2} \arctan \left (\frac{b x^{2} - 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^3),x, algorithm="fricas")

[Out]

[1/8*(b*x^2*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)*sqrt(-a))/(sqr
t(-a)*a*x^2), 1/4*(b*x^2*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)*sqrt(a))/(a^(3/2)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(-a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [A]  time = 0.324625, size = 93, normalized size = 1.21 \[ \frac{b{\rm ln}\left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{4 \, \sqrt{-a} a} + \frac{\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^3),x, algorithm="giac")

[Out]

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