Optimal. Leaf size=74 \[ \frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0403233, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x**2+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0161419, size = 36, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a x+b x^3\right )}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 33, normalized size = 0.5 \[{\frac{x \left ( b{x}^{2}+3\,a \right ) }{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x^2+a)^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.703227, size = 14, normalized size = 0.19 \[ \frac{1}{3} \, b x^{3} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.254874, size = 14, normalized size = 0.19 \[ \frac{1}{3} \, b x^{3} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.153532, size = 8, normalized size = 0.11 \[ a x + \frac{b x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x**2+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.268615, size = 27, normalized size = 0.36 \[ \frac{1}{3} \,{\left (b x^{3} + 3 \, a x\right )}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2),x, algorithm="giac")
[Out]