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

Optimal. Leaf size=72 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^2+c x^4}}{2 a x^2} \]

[Out]

-Sqrt[a + b*x^2 + c*x^4]/(2*a*x^2) + (b*ArcTanh[(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 = 0.127957, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^2+c x^4}}{2 a x^2} \]

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*ArcTanh[(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 = 14.6344, size = 61, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{2} + c x^{4}}}{2 a x^{2}} + \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 a^{\frac{3}{2}}} \]

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*atanh((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 = 0.123607, size = 78, normalized size = 1.08 \[ -\frac{b \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^2+c x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.016, size = 63, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{b}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

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^(3/2)*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 = 0.298947, size = 1, normalized size = 0.01 \[ \left [\frac{b x^{2} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{2}}, \frac{b x^{2} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{4 \, \sqrt{-a} a x^{2}}\right ] \]

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))/(a^(3
/2)*x^2), 1/4*(b*x^2*arctan(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*
a)) - 2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a} x^{3}}\,{d x} \]

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]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x^3), x)