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

Optimal. Leaf size=50 \[ \frac{\sqrt{b x^2+c x^4}}{x}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0900042, antiderivative size = 50, 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 \[ \frac{\sqrt{b x^2+c x^4}}{x}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9791, size = 41, normalized size = 0.82 \[ - \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )} + \frac{\sqrt{b x^{2} + c x^{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0711607, size = 75, normalized size = 1.5 \[ \frac{x \sqrt{b+c x^2} \left (\sqrt{b+c x^2}-\sqrt{b} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b} \log (x)\right )}{\sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[b + c*x^2]*(Sqrt[b + c*x^2] + Sqrt[b]*Log[x] - Sqrt[b]*Log[b + Sqrt[b]*S
qrt[b + c*x^2]]))/Sqrt[x^2*(b + c*x^2)]

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Maple [A]  time = 0.008, size = 65, normalized size = 1.3 \[ -{\frac{1}{x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) -\sqrt{c{x}^{2}+b} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^2,x)

[Out]

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

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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(sqrt(c*x^4 + b*x^2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278106, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} x \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}}{2 \, x}, -\frac{\sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}\right ) - \sqrt{c x^{4} + b x^{2}}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275216, size = 92, normalized size = 1.84 \[{\left (\frac{b \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \sqrt{c x^{2} + b}\right )}{\rm sign}\left (x\right ) - \frac{{\left (b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b} \sqrt{b}\right )}{\rm sign}\left (x\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*arctan(sqrt(c*x^2 + b)/sqrt(-b))/sqrt(-b) + sqrt(c*x^2 + b))*sign(x) - (b*arc
tan(sqrt(b)/sqrt(-b)) + sqrt(-b)*sqrt(b))*sign(x)/sqrt(-b)