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

Optimal. Leaf size=59 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0965936, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.587, size = 49, normalized size = 0.83 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{2 b x^{3}} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0728432, size = 97, normalized size = 1.64 \[ \frac{-\sqrt{b} \left (b+c x^2\right )-c x^2 \log (x) \sqrt{b+c x^2}+c x^2 \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{2 b^{3/2} x \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 73, normalized size = 1.2 \[{\frac{1}{2\,x}\sqrt{c{x}^{2}+b} \left ( c\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}b-\sqrt{c{x}^{2}+b}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/x*(c*x^2+b)^(1/2)*(c*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^2*b-(c*x^2+b)^(1/
2)*b^(3/2))/(c*x^4+b*x^2)^(1/2)/b^(5/2)

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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)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276566, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} c x^{3} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} b}{4 \, b^{2} x^{3}}, -\frac{\sqrt{-b} c x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} b}{2 \, b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(sqrt(b)*c*x^3*log(-((c*x^3 + 2*b*x)*sqrt(b) + 2*sqrt(c*x^4 + b*x^2)*b)/x^3
) - 2*sqrt(c*x^4 + b*x^2)*b)/(b^2*x^3), -1/2*(sqrt(-b)*c*x^3*arctan(sqrt(-b)*x/s
qrt(c*x^4 + b*x^2)) + sqrt(c*x^4 + b*x^2)*b)/(b^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError