Optimal. Leaf size=22 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.00918767, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^2]/Sqrt[1 + x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.45317, size = 17, normalized size = 0.77 \[ \frac{\sqrt{a x^{2}} \sqrt{x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**2)**(1/2)/(x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0079487, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*x^2]/Sqrt[1 + x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 19, normalized size = 0.9 \[{\frac{1}{x}\sqrt{a{x}^{2}}\sqrt{{x}^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^2)^(1/2)/(x^2+1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.768608, size = 26, normalized size = 1.18 \[ \frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.290682, size = 74, normalized size = 3.36 \[ -\frac{\sqrt{a x^{2}}{\left (2 \, x^{3} -{\left (2 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 2 \, x\right )}}{2 \, x^{3} - 2 \, \sqrt{x^{2} + 1} x^{2} + x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.11581, size = 20, normalized size = 0.91 \[ \frac{\sqrt{a} \sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**2)**(1/2)/(x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.264582, size = 26, normalized size = 1.18 \[{\left (\sqrt{x^{2} + 1}{\rm sign}\left (x\right ) -{\rm sign}\left (x\right )\right )} \sqrt{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="giac")
[Out]