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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]