3.134 \(\int \frac{1}{\sqrt{x^2 \left (a+b x^2+c x^4\right )}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right )}{2 \sqrt{a}} \]

[Out]

-ArcTanh[(x*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^4 + c*x^6])]/(2*Sqrt[a])

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Rubi [A]  time = 0.0321583, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[(x*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^4 + c*x^6])]/(2*Sqrt[a])

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Rubi in Sympy [A]  time = 26.1205, size = 75, normalized size = 1.53 \[ - \frac{x \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a} \sqrt{a x^{2} + b x^{4} + c x^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(a + b*x**2 + c*x**4)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c
*x**4)))/(2*sqrt(a)*sqrt(a*x**2 + b*x**4 + c*x**6))

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Mathematica [A]  time = 0.0796575, size = 87, normalized size = 1.78 \[ \frac{x \sqrt{a+b x^2+c x^4} \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )\right )}{2 \sqrt{a} \sqrt{x^2 \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 72, normalized size = 1.5 \[ -{\frac{x}{2}\sqrt{c{x}^{4}+b{x}^{2}+a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{{x}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299648, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \, \sqrt{c x^{6} + b x^{4} + a x^{2}}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{5}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{{\left (b x^{3} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{4} + a x^{2}} a}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*log((4*sqrt(c*x^6 + b*x^4 + a*x^2)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^5 +
8*a*b*x^3 + 8*a^2*x)*sqrt(a))/x^5)/sqrt(a), 1/2*sqrt(-a)*arctan(1/2*(b*x^3 + 2*
a*x)*sqrt(-a)/(sqrt(c*x^6 + b*x^4 + a*x^2)*a))/a]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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